#7094726

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, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1235 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=ghi,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=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, 7001922);