#7095257

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

1317 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)(19,31)(20,35)(21,27)(22,29)(23,28)(24,30)(25,34)(26,38)(32,45)(33,41)(36,43)(37,44)(39,46)(40,48)(42,47);(1,3,5,4)(2,6,8,7)(9,10)(11,15)(12,14)(13,16)(17,19,23,34,24,21,22,33)(18,25,29,31,30,41,28,27)(20,36,37,48,38,32,39,47)(26,42,44,45,35,40,46,43);(1,3,5,4)(2,7,8,6)(9,11,12,16)(10,13,14,15)(17,20)(18,26)(19,32)(21,36)(22,39)(23,37)(24,38)(25,40)(27,45)(28,46)(29,44)(30,35)(31,43)(33,48)(34,47)(41,42);(11,16)(13,15)(17,21,24,19)(18,27,30,31)(20,32,38,36)(22,33,23,34)(25,29,41,28)(26,43,35,45)(37,48,39,47)(40,46,42,44);(1,4,5,3)(2,7,8,6)(10,14)(13,15)(17,22,24,23)(18,28,30,29)(19,33,21,34)(20,37,38,39)(25,27,41,31)(26,44,35,46)(32,47,36,48)(40,43,42,45)

Polycyclic presentation:

2^9|a^2=gi,b^2=fgi,c^2=gh,d^2=i,e^2=g,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, 7002453);