#7012682

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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

1416 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)(17,18,24,28,23,30,22,29)(19,25,32,45,34,42,33,41)(20,27,36,26,21,38,37,44)(31,46,48,39,35,40,47,43);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,19)(18,25)(20,35)(21,31)(22,33)(23,34)(24,32)(26,43)(27,40)(28,45)(29,41)(30,42)(36,47)(37,48)(38,46)(39,44);(1,3,5,4)(2,7,6,8)(9,11)(10,13)(12,16)(14,15)(17,20)(18,26)(19,31)(21,23)(22,36)(24,37)(25,39)(27,28)(29,38)(30,44)(32,47)(33,48)(34,35)(40,41)(42,43)(45,46);(2,6)(7,8)(11,16)(13,15)(17,21,23,20)(18,27,30,38)(19,31,34,35)(22,36,24,37)(25,40,42,46)(26,29,44,28)(32,48,33,47)(39,45,43,41);(1,4,5,3)(2,8,6,7)(10,14)(13,15)(17,22,23,24)(18,28,30,29)(19,32,34,33)(20,36,21,37)(25,41,42,45)(26,27,44,38)(31,47,35,48)(39,40,43,46)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6919878);