#6859381

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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	#2414	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1474 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)(17,18,22,27)(19,28,23,24)(20,31,33,42)(21,34,35,26)(25,37,40,29)(30,38,36,41)(32,45,46,44)(39,43,47,48);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,16)(13,14)(17,19,22,23)(18,24,27,28)(20,29,33,37)(21,30,35,36)(25,31,40,42)(26,38,34,41)(32,43,46,48)(39,44,47,45);(10,14)(15,16)(17,20,22,33)(18,25,27,40)(19,29,23,37)(21,32,35,46)(24,31,28,42)(26,39,34,47)(30,43,36,48)(38,44,41,45);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(17,21)(18,26)(19,30)(20,32)(22,35)(23,36)(24,38)(25,39)(27,34)(28,41)(29,43)(31,44)(33,46)(37,48)(40,47)(42,45);(3,8)(5,7)(9,12,13,11)(10,15,14,16)(20,33)(21,35)(25,40)(26,34)(29,37)(30,36)(31,42)(38,41)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=h,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,e^c=ei,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, 6766577);