#6910886

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907993
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1450 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)(3,4)(17,18,22,27)(19,26,30,23)(20,31,33,41)(21,34,37,25)(24,39,42,28)(29,38,35,43)(32,48,36,47)(40,45,44,46);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,19,22,30)(18,23,27,26)(20,28,33,39)(21,35,37,29)(24,31,42,41)(25,43,34,38)(32,46,36,45)(40,48,44,47);(1,3)(2,7)(4,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,20,19,28,22,33,30,39)(18,24,23,31,27,42,26,41)(21,36,29,46,37,32,35,45)(25,44,38,48,34,40,43,47);(1,4,5,3)(2,8,6,7)(11,16)(13,15)(17,21,22,37)(18,25,27,34)(19,29,30,35)(20,32,33,36)(23,38,26,43)(24,40,42,44)(28,45,39,46)(31,47,41,48);(2,6)(7,8)(10,14)(13,15)(17,19,22,30)(18,26,27,23)(20,28,33,39)(21,35,37,29)(24,41,42,31)(25,38,34,43)(32,46,36,45)(40,47,44,48)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fgi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,d^c=dfh,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, 6818082);