#6366800

order	512
nilpotency class	3
isoclinism	isoclinic class of #6366710
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
prime radical	#3511	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1465 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6273996);