#7090497

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1683 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)(19,29)(20,25)(21,26)(22,37)(23,27)(24,31)(28,35)(30,41)(32,44)(33,43)(34,48)(36,39)(38,46)(40,47)(42,45);(9,10)(11,15)(12,14)(13,16)(17,19,22,31,23,21,35,36)(18,24,28,29,27,39,37,26)(20,32,34,47,33,30,45,46)(25,40,42,44,43,38,48,41);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,20,23,33)(18,25,27,43)(19,30,21,32)(22,34,35,45)(24,38,39,40)(26,44,29,41)(28,42,37,48)(31,46,36,47);(11,16)(13,15)(17,21,23,19)(18,26,27,29)(20,30,33,32)(22,31,35,36)(24,28,39,37)(25,41,43,44)(34,47,45,46)(38,48,40,42);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(18,27)(19,21)(20,33)(28,37)(31,36)(34,45)(38,40)(41,44)

Polycyclic presentation:

2^9|b^2=f,c^2=gh,d^2=i,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,e^c=ei,e^d=ei,f^d=fi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6997693);