#6994793

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	#42992	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1775 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,e^c=ei,g^c=gi,e^d=ei

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, 6901989);