#6367320

order	512
nilpotency class	3
isoclinism	isoclinic class of #6367198
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1499 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,f^b=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306988, #72307013, #72307019
maximal normal subgroup of	#72306988, #72307013, #72307019

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6274516);