#7273197

order	512
nilpotency class	3
isoclinism	isoclinic class of #7234345
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	#42761	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1443 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)(3,6)(4,5)(7,8)(11,15)(13,16)(17,18,23,27,22,29,21,28)(19,24,36,39,32,41,31,40)(20,26,33,25,35,43,34,42)(30,38,44,37,46,48,45,47);(2,5)(3,7)(9,10)(11,16)(12,14)(13,15)(17,19,22,32)(18,24,29,41)(20,30,35,46)(21,31,23,36)(25,37,42,47)(26,38,43,48)(27,39,28,40)(33,44,34,45);(9,11,12,15)(10,13,14,16)(17,20)(18,25)(19,30)(21,33)(22,35)(23,34)(24,37)(26,27)(28,43)(29,42)(31,44)(32,46)(36,45)(38,39)(40,48)(41,47);(1,3,4,7)(2,6,5,8)(11,15)(13,16)(17,20)(18,26)(19,30)(21,34)(22,35)(23,33)(24,38)(25,27)(28,42)(29,43)(31,45)(32,46)(36,44)(37,39)(40,47)(41,48);(2,5)(6,8)(10,14)(13,16)(17,21,22,23)(18,27,29,28)(19,31,32,36)(20,33,35,34)(24,39,41,40)(25,26,42,43)(30,44,46,45)(37,38,47,48)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7180393);