#7092190

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
prime radical	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1356 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,c^2=ghi,d^2=i,e^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=ehi,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, 6999386);