#72319611

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

2446 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)(4,8)(5,10)(6,11)(9,13)(12,18)(14,19)(15,21)(16,22)(17,24)(20,25)(23,27)(26,29)(28,31)(30,32)(33,34)(35,44)(36,48)(37,42)(38,41)(39,43)(40,46)(45,54)(47,58)(49,53)(50,62)(51,57)(52,56)(55,60)(59,64)(61,63);(1,3)(2,7)(4,12)(5,14)(6,13)(8,18)(9,11)(10,19)(15,26)(16,25)(17,27)(20,22)(21,29)(23,24)(28,32)(30,31)(33,35,37,47,38,45,52,59,39,46,51,60,36,49,50,61)(34,40,42,55,41,53,56,63,43,44,57,58,48,54,62,64);(1,4,5,15,6,16,17,28)(2,8,10,21,11,22,24,31)(3,12,14,26,13,25,27,32)(7,18,19,29,9,20,23,30)(35,45,46,49)(37,51)(40,53,44,54)(42,57)(47,61,60,59)(50,52)(55,64,58,63)(56,62);(1,3)(2,9)(4,12)(5,14)(6,13)(7,11)(8,20)(10,23)(15,26)(16,25)(17,27)(18,22)(19,24)(21,30)(28,32)(29,31)(33,36,39,38)(34,41,43,48)(35,45,46,49)(37,50,51,52)(40,54,44,53)(42,56,57,62)(47,59,60,61)(55,64,58,63);(3,13)(7,9)(12,25)(14,27)(18,20)(19,23)(26,32)(29,30)(35,46)(40,44)(45,49)(47,60)(53,54)(55,58)(59,61)(63,64)

Polycyclic presentation:

2^10|b^2=g,c^2=f,d^2=j,f^2=h,g^2=i,i^2=j,b^a=bj,c^b=cij,d^a=dhj,d^b=dj,e^b=ehj,e^d=eh,f^b=fj,g^c=gj

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319631, #72319792, #72343221
maximal normal subgroup of	#72319631, #72319792, #72343221

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