#7089870

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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1434 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)(19,25)(20,26)(21,28)(22,27)(23,29)(24,31)(30,40)(32,41)(33,48)(34,45)(35,44)(36,39)(37,43)(38,46)(42,47);(9,10)(11,15)(12,14)(13,16)(17,19,23,32)(18,24,29,39)(20,33,35,46)(21,31,22,36)(25,28,41,27)(26,42,44,40)(30,37,47,34)(38,45,48,43);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,19)(18,25)(20,30)(21,36)(22,31)(23,32)(24,27)(26,40)(28,39)(29,41)(33,34)(35,47)(37,46)(38,43)(42,44)(45,48);(11,16)(13,15)(17,20,23,35)(18,26,29,44)(19,30,32,47)(21,34,22,37)(24,38,39,48)(25,40,41,42)(27,43,28,45)(31,46,36,33);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(17,21,23,22)(18,27,29,28)(19,31,32,36)(20,34,35,37)(24,25,39,41)(26,43,44,45)(30,46,47,33)(38,40,48,42)

Polycyclic presentation:

2^9|b^2=i,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=df,e^b=ehi,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, 6997066);