#6993248

order	512
nilpotency class	3
isoclinism	isoclinic class of #6885129
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	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1549 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,7)(4,6)(5,8)(17,18,23,26,22,28,21,27)(19,24,36,39,30,38,31,40)(20,32,33,25,35,42,34,41)(29,43,45,37,44,48,46,47);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,15)(13,16)(17,19)(18,24)(20,29)(21,31)(22,30)(23,36)(25,37)(26,39)(27,40)(28,38)(32,43)(33,45)(34,46)(35,44)(41,47)(42,48);(3,8)(5,7)(9,11,14,10)(12,15,13,16)(17,20)(18,25)(19,29)(21,33)(22,35)(23,34)(24,37)(26,32)(27,42)(28,41)(30,44)(31,45)(36,46)(38,47)(39,43)(40,48);(2,6)(5,7)(9,12,14,13)(10,15,11,16)(19,30)(24,38)(29,44)(31,36)(37,47)(39,40)(43,48)(45,46);(9,13,14,12)(10,15,11,16)(17,21,22,23)(18,26,28,27)(19,31,30,36)(20,33,35,34)(24,39,38,40)(25,32,41,42)(29,45,44,46)(37,43,47,48)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6900444);