#7017098

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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1612 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:

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

Polycyclic presentation:

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

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, 6924294);