#7055284

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	#45092	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1744 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)(19,27)(20,31)(21,26)(22,28)(23,24)(25,29)(30,42)(32,47)(33,40)(34,38)(35,48)(36,44)(37,41)(39,45)(43,46);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(17,19,22,23)(18,24,28,27)(20,29,33,37)(21,34,36,30)(25,31,41,40)(26,42,44,38)(32,46,35,45)(39,48,43,47);(9,11,12,16)(10,13,14,15)(17,20)(18,25)(19,29)(21,35)(22,33)(23,37)(24,31)(26,43)(27,40)(28,41)(30,46)(32,36)(34,45)(38,48)(39,44)(42,47);(3,8)(5,7)(11,16)(13,15)(17,21)(18,26)(19,30)(20,32)(22,36)(23,34)(24,38)(25,39)(27,42)(28,44)(29,45)(31,47)(33,35)(37,46)(40,48)(41,43);(10,14)(13,15)(17,19,22,23)(18,27,28,24)(20,29,33,37)(21,34,36,30)(25,40,41,31)(26,38,44,42)(32,46,35,45)(39,47,43,48)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=h,e^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,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, 6962480);