#6929973

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907993
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	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1162 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fi,c^2=fgi,d^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,d^c=dfh,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, 6837169);