#6858760

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1464 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)(3,7)(4,6)(5,8)(17,18,22,27)(19,28,23,24)(20,31,34,39)(21,35,36,26)(25,32,41,29)(30,38,37,42)(33,46,47,44)(40,43,48,45);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,16)(13,14)(17,19,22,23)(18,24,27,28)(20,32,34,29)(21,30,36,37)(25,39,41,31)(26,38,35,42)(33,45,47,43)(40,46,48,44);(10,14)(15,16)(17,20,22,34)(18,25,27,41)(19,29,23,32)(21,33,36,47)(24,31,28,39)(26,40,35,48)(30,43,37,45)(38,44,42,46);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(17,21)(18,26)(19,30)(20,33)(22,36)(23,37)(24,38)(25,40)(27,35)(28,42)(29,43)(31,44)(32,45)(34,47)(39,46)(41,48);(3,8)(5,7)(9,12,13,11)(10,15,14,16)(20,34)(21,36)(25,41)(26,35)(29,32)(30,37)(31,39)(38,42)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=hi,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6765956);