#7012757

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	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1417 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=fi,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=ehi,d^c=dh,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, 6919953);