#6948026

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884841
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1156 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)(3,11)(4,5)(6,9)(7,10)(8,14)(12,15)(13,16)(17,18)(19,22)(20,24)(21,23)(25,29)(26,28)(27,30)(31,32)(33,34,39,52,38,44,37,43)(35,40,53,63,48,57,47,56)(36,42,49,41,51,58,50,59)(45,60,61,54,46,55,62,64);(1,3)(2,8)(4,12)(5,13)(6,15)(7,14)(9,16)(10,11)(17,19)(18,22)(20,26)(21,25)(23,29)(24,28)(27,31)(30,32)(33,35)(34,40)(36,45)(37,47)(38,48)(39,53)(41,54)(42,60)(43,56)(44,57)(46,51)(49,61)(50,62)(52,63)(55,58)(59,64);(1,4,7,9)(2,5,10,6)(3,12,14,16)(8,13,11,15)(18,23)(19,25)(20,27)(28,32)(33,36)(34,41)(35,45)(37,49)(38,51)(39,50)(40,54)(42,52)(43,58)(44,59)(46,48)(47,61)(53,62)(55,56)(57,64)(60,63);(1,5,7,6)(2,9,10,4)(3,13,14,15)(8,16,11,12)(17,20,21,27)(18,24,23,30)(19,26,25,31)(22,28,29,32)(33,36)(34,42)(35,46)(37,50)(38,51)(39,49)(40,55)(41,52)(43,59)(44,58)(45,48)(47,61)(53,62)(54,56)(57,60)(63,64);(1,6,7,5)(2,9,10,4)(3,13,14,15)(8,12,11,16)(19,25)(22,29)(26,31)(28,32)(33,37,38,39)(34,43,44,52)(35,47,48,53)(36,49,51,50)(40,56,57,63)(41,58,59,42)(45,61,46,62)(54,55,64,60)

Polycyclic presentation:

2^9|a^2=ghi,c^2=f,d^2=fh,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,e^b=efh,d^c=dfh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6855222);