#6631999

order	512
nilpotency class	3
isoclinism	isoclinic class of #6631116
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1137 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)(5,6)(9,12)(10,14)(11,13)(15,16)(17,18,21,24)(19,22,26,30)(20,23,28,32)(25,29,27,31)(33,34,39,45,40,46,38,44)(35,41,49,57,50,58,54,60)(36,51,52,59,37,43,53,42)(47,61,63,64,48,56,62,55);(2,6)(7,12)(8,13)(14,16)(17,19,21,26)(18,22,24,30)(20,27,28,25)(23,31,32,29)(33,35)(34,41)(36,47)(37,48)(38,54)(39,49)(40,50)(42,55)(43,56)(44,60)(45,57)(46,58)(51,61)(52,63)(53,62)(59,64);(1,3)(2,7)(4,10)(5,9)(6,12)(8,14)(11,15)(13,16)(17,20)(18,23)(19,25)(21,28)(22,29)(24,32)(26,27)(30,31)(33,36,40,37)(34,42,46,59)(35,47,50,48)(38,52,39,53)(41,55,58,64)(43,44,51,45)(49,62,54,63)(56,60,61,57);(3,9)(7,12)(10,15)(14,16)(18,24)(20,28)(22,30)(25,27)(33,37,40,36)(34,43,46,51)(35,48,50,47)(38,52,39,53)(41,56,58,61)(42,44,59,45)(49,62,54,63)(55,60,64,57);(1,4,5,11)(2,8,6,13)(3,10,9,15)(7,14,12,16)(33,38,40,39)(34,44,46,45)(35,49,50,54)(36,52,37,53)(41,57,58,60)(42,51,59,43)(47,62,48,63)(55,56,64,61)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=i,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dh,c^b=ch,e^b=ei,d^c=dfh,g^c=gi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6539195);