#7094602

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

1279 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=gh,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,f^b=fi,d^c=dh,e^c=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, 7001798);