#6867884

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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1501 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)(17,18,23,25,21,27,22,28)(19,29,36,38,30,24,32,40)(20,33,37,42,34,26,35,41)(31,39,46,48,44,43,45,47);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,16)(13,14)(17,19)(18,24)(20,31)(21,30)(22,32)(23,36)(25,40)(26,39)(27,29)(28,38)(33,43)(34,44)(35,45)(37,46)(41,48)(42,47);(10,14)(15,16)(18,25)(19,30)(22,23)(24,38)(26,41)(27,28)(29,40)(31,44)(33,42)(35,37)(39,47)(43,48);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(17,20)(18,26)(19,31)(21,34)(22,35)(23,37)(24,39)(25,41)(27,33)(28,42)(29,43)(30,44)(32,45)(36,46)(38,47)(40,48);(3,8)(5,7)(9,12,13,11)(10,15,14,16)(20,34)(26,33)(31,44)(35,37)(39,43)(41,42)(45,46)(47,48)

Polycyclic presentation:

2^9|a^2=fg,d^2=hi,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,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, 6775080);