#7090074

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	#2414	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1443 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,24,29)(19,30,34,42)(20,26,21,27)(22,28,23,38)(25,39,43,33)(31,47,36,41)(32,46,48,40)(35,44,37,45);(9,10)(11,15)(12,14)(13,16)(17,19,24,34)(18,25,29,43)(20,31,21,36)(22,33,23,39)(26,40,27,46)(28,42,38,30)(32,37,48,35)(41,44,47,45);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,20)(18,26)(19,31)(21,24)(22,37)(23,35)(25,40)(27,29)(28,45)(30,47)(32,39)(33,48)(34,36)(38,44)(41,42)(43,46);(11,16)(13,15)(17,21,24,20)(18,27,29,26)(19,32,34,48)(22,37,23,35)(25,41,43,47)(28,45,38,44)(30,46,42,40)(31,33,36,39);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(17,22,24,23)(18,28,29,38)(19,33,34,39)(20,35,21,37)(25,42,43,30)(26,44,27,45)(31,32,36,48)(40,47,46,41)

Polycyclic presentation:

2^9|a^2=i,b^2=i,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, 6997270);