#7095202

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1310 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)(19,30)(20,26)(21,27)(22,28)(23,38)(24,29)(25,32)(31,45)(33,41)(34,42)(35,44)(36,40)(37,43)(39,47)(46,48);(1,3,5,4)(2,6,8,7)(9,10)(11,15)(12,14)(13,16)(17,19,23,32,24,21,22,36)(18,25,28,30,29,40,38,27)(20,31,34,47,35,33,37,46)(26,39,43,45,44,48,42,41);(1,3,5,4)(2,7,8,6)(9,11,12,16)(10,13,14,15)(17,20)(18,26)(19,31)(21,33)(22,37)(23,34)(24,35)(25,39)(27,41)(28,43)(29,44)(30,45)(32,47)(36,46)(38,42)(40,48);(11,16)(13,15)(17,21,24,19)(18,27,29,30)(20,33,35,31)(22,36,23,32)(25,28,40,38)(26,41,44,45)(34,47,37,46)(39,43,48,42);(1,4,5,3)(2,7,8,6)(10,14)(13,15)(17,22,24,23)(18,28,29,38)(19,32,21,36)(20,34,35,37)(25,27,40,30)(26,42,44,43)(31,46,33,47)(39,45,48,41)

Polycyclic presentation:

2^9|a^2=g,b^2=fg,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,d^c=dh,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, 7002398);