#7094682

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

1258 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)(17,18,24,30)(19,31,36,42)(20,26,21,27)(22,29,23,28)(25,34,43,35)(32,41,37,48)(33,40,47,44)(38,46,39,45);(1,3,5,4)(2,6,8,7)(9,10)(11,15)(12,14)(13,16)(17,19,24,36)(18,25,30,43)(20,37,21,32)(22,35,23,34)(26,44,27,40)(28,31,29,42)(33,39,47,38)(41,46,48,45);(1,3,5,4)(2,7,8,6)(9,11,12,16)(10,13,14,15)(17,20,24,21)(18,26,30,27)(19,32,36,37)(22,39,23,38)(25,40,43,44)(28,46,29,45)(31,41,42,48)(33,34,47,35);(11,16)(13,15)(17,21,24,20)(18,27,30,26)(19,33,36,47)(22,39,23,38)(25,41,43,48)(28,46,29,45)(31,40,42,44)(32,34,37,35);(1,4,5,3)(2,7,8,6)(10,14)(13,15)(17,22,24,23)(18,28,30,29)(19,34,36,35)(20,38,21,39)(25,42,43,31)(26,45,27,46)(32,33,37,47)(40,41,44,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,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=eh,f^b=fi,d^c=dhi,e^c=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, 7001878);