#6651180

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632074
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

1465 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=hi,d^2=h,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=dfh,e^c=ei,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, 6558376);