#6651300

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632088
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	#2540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1467 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)(17,18,22,28)(19,29,30,24)(20,33,35,26)(21,27,23,25)(31,39,44,43)(32,38,36,40)(34,42,37,41)(45,47,46,48);(1,2)(3,6)(4,5)(7,8)(11,12)(14,15)(17,19,22,30)(18,24,28,29)(20,31,35,44)(21,36,23,32)(25,40,27,38)(26,39,33,43)(34,46,37,45)(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)(33,42)(34,37)(39,47)(43,48);(2,5)(6,7)(9,12,13,11)(10,15,16,14)(17,20,22,35)(18,26,28,33)(19,31,30,44)(21,37,23,34)(24,39,29,43)(25,41,27,42)(32,46,36,45)(38,47,40,48);(1,3,4,8)(2,6,5,7)(10,16)(14,15)(17,21,22,23)(18,27,28,25)(19,32,30,36)(20,34,35,37)(24,40,29,38)(26,41,33,42)(31,45,44,46)(39,47,43,48)

Polycyclic presentation:

2^9|a^2=f,b^2=i,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^b=ei,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, 6558496);