#7080358

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1477 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=hi,d^2=gi,e^2=i,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=dhi,e^c=ei,f^d=fi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6987554);