#6631979

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dh,c^b=ch,d^b=di,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, 6539175);