#6651219

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632048
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	#30512	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1474 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)(19,27)(20,31)(21,25)(22,36)(23,29)(24,34)(26,35)(28,44)(30,39)(32,48)(33,42)(37,41)(38,47)(40,45)(43,46);(1,2)(3,6)(4,5)(7,8)(11,12)(14,15)(17,19,21,29)(18,23,25,27)(20,32,33,28)(22,30,35,46)(24,40,41,38)(26,39,36,43)(31,44,42,48)(34,47,37,45);(9,11,13,12)(10,14,16,15)(17,20,21,33)(18,24,25,41)(19,28,29,32)(22,37,35,34)(23,38,27,40)(26,42,36,31)(30,47,46,45)(39,44,43,48);(2,5)(6,7)(9,12,13,11)(10,15,16,14)(18,25)(23,27)(24,41)(26,36)(31,42)(38,40)(39,43)(44,48);(1,3,4,8)(2,6,5,7)(10,16)(14,15)(19,29)(23,27)(28,32)(30,46)(38,40)(39,43)(44,48)(45,47)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=h,d^2=hi,e^2=fi,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,g^c=gi

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, 6558415);