#7091144

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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×#547	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1586 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,22,27,23,28,38,41)(19,29,32,26,31,45,21,24)(20,33,37,48,36,25,42,44)(30,47,35,40,34,46,39,43);(9,10)(11,15)(12,14)(13,16)(17,19,23,31)(18,24,28,26)(20,34,36,30)(21,38,32,22)(25,43,33,40)(27,45,41,29)(35,42,39,37)(44,46,48,47);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,20)(18,25)(19,30)(21,39)(22,37)(23,36)(24,40)(26,43)(27,44)(28,33)(29,46)(31,34)(32,35)(38,42)(41,48)(45,47);(11,16)(13,15)(17,21,22,19,23,32,38,31)(18,26,27,45,28,24,41,29)(20,35,37,34,36,39,42,30)(25,40,44,46,33,43,48,47);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(19,31)(20,36)(21,32)(24,26)(25,33)(29,45)(37,42)(44,48)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gh,d^2=f,e^2=gi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6998340);