#6650275

order	512
nilpotency class	3
isoclinism	isoclinic class of #6636940
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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1128 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)(3,10)(4,8)(5,14)(6,9)(7,12)(11,16)(13,15)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,31)(28,32)(33,34,38,43)(35,42,37,40)(36,47,48,63)(39,53,52,44)(41,56,57,49)(45,60,50,62)(46,55,51,59)(54,61,58,64);(1,3)(2,7)(4,11)(5,13)(6,12)(8,15)(9,10)(14,16)(17,19,21,25)(18,22,24,29)(20,27,28,26)(23,31,32,30)(33,35)(34,40)(36,45)(37,38)(39,46)(41,54)(42,43)(44,55)(47,62)(48,50)(49,61)(51,52)(53,59)(56,64)(57,58)(60,63);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,10,16)(19,25)(22,29)(26,27)(30,31)(33,36,39,49,38,48,52,56)(34,41,44,47,43,57,53,63)(35,45,46,61,37,50,51,64)(40,54,55,62,42,58,59,60);(2,9)(3,12)(8,14)(11,13)(17,20,21,28)(18,23,24,32)(19,26,25,27)(22,30,29,31)(33,37,38,35)(34,42,43,40)(36,45,48,50)(39,51,52,46)(41,54,57,58)(44,59,53,55)(47,62,63,60)(49,61,56,64);(1,5,6,4)(2,8,9,14)(3,13,12,11)(7,15,10,16)(18,24)(22,29)(23,32)(30,31)(34,43)(35,37)(36,48)(44,53)(46,51)(49,56)(54,58)(60,62)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=fgi,d^2=hi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=dfh,e^b=ei,d^c=di,e^c=ei,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, 6557471);