#72287536

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator	C₂×C₄	C₂³×C₄²
Frattini, cosocle	C₄×C₈	C₂⁵
socle	C₂²	C₂³×#97
prime radical	#3444	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2448 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 64 points:

(1,2)(3,7)(4,10)(5,11)(6,9)(8,12)(13,18)(14,19)(15,24)(16,22)(17,23)(20,25)(21,26)(27,29)(28,31)(30,32)(33,34)(35,47)(36,42)(37,43)(38,44)(39,45)(40,46)(41,50)(48,61)(49,59)(51,58)(52,57)(53,60)(54,62)(55,63)(56,64);(1,3)(2,7)(4,13)(5,14)(6,12)(8,9)(10,18)(11,19)(15,27)(16,25)(17,26)(20,22)(21,23)(24,29)(28,32)(30,31);(1,3)(2,8)(4,13)(5,14)(6,12)(7,9)(10,20)(11,21)(15,27)(16,25)(17,26)(18,22)(19,23)(24,30)(28,32)(29,31)(33,35,39,48,38,51,54,53,40,50,56,52,55,49,37,36)(34,41,45,57,44,59,62,42,46,47,64,61,63,58,43,60);(2,9)(3,12)(10,22)(11,23)(13,25)(14,26)(24,31)(27,32)(33,36,37,49,55,52,56,50,40,53,54,51,38,48,39,35)(34,42,43,59,63,57,64,41,46,60,62,58,44,61,45,47);(1,4,5,15,6,16,17,28)(2,10,11,24,9,22,23,31)(3,13,14,27,12,25,26,32)(7,18,19,29,8,20,21,30)(33,37,38,39,40,54,55,56)(34,43,44,45,46,62,63,64)(35,48,49,36,50,52,51,53)(41,57,58,60,47,61,59,42)

Polycyclic presentation:

2^10|c^2=g,d^2=gij,e^2=f,f^2=h,g^2=i,i^2=j,c^a=ch,d^a=dhj,d^b=dhj,d^c=dhj,e^c=eij,e^d=ei,f^c=fj,f^d=fj,g^e=gj

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287605, #72287851, #72319792, …
maximal normal subgroup of	#72287605, #72287851, #72319792, …

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 3, F.7);;
gap> SetPower(coll, 4, F.7*F.9*F.10);;
gap> SetPower(coll, 5, F.6);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.8*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);