#72287632

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	C₂²×#41	M₁₆
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2524 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:

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

Polycyclic presentation:

2^10|a^2=f,c^2=g,e^2=gi,f^2=h,g^2=i,i^2=j,c^a=cij,d^c=dhj,e^a=eij,e^b=eh,e^d=ej,f^c=fj,f^e=fj,g^a=gj

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287657, #72287751, #72320746
maximal normal subgroup of	#72287657, #72287751, #72320746

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, 1, F.6);;
gap> SetPower(coll, 3, F.7);;
gap> SetPower(coll, 5, F.7*F.9);;
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.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);