#6631992

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

1141 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)(33,34,40,55,39,45,38,44)(35,46,48,59,49,41,50,58)(36,43,52,42,37,54,53,61)(47,57,64,60,51,62,63,56);(2,6)(7,12)(8,13)(14,16)(17,19,21,26)(18,22,24,30)(20,27,28,25)(23,31,32,29)(33,35,39,49)(34,41,45,46)(36,51,37,47)(38,50,40,48)(42,60,61,56)(43,57,54,62)(44,59,55,58)(52,63,53,64);(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)(33,36,39,37)(34,42,45,61)(35,47,49,51)(38,52,40,53)(41,56,46,60)(43,44,54,55)(48,63,50,64)(57,58,62,59);(3,9)(7,12)(10,15)(14,16)(18,24)(20,28)(22,30)(25,27)(33,37,39,36)(34,43,45,54)(35,47,49,51)(38,52,40,53)(41,57,46,62)(42,44,61,55)(48,64,50,63)(56,58,60,59);(1,4,5,11)(2,8,6,13)(3,10,9,15)(7,14,12,16)(33,38,39,40)(34,44,45,55)(35,48,49,50)(36,52,37,53)(41,58,46,59)(42,54,61,43)(47,63,51,64)(56,62,60,57)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=i,d^2=i,e^2=f,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, 6539188);