#6820715

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806647
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	#33187	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1580 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)(3,6)(4,5)(7,8)(9,10,13,15)(11,14,12,16)(19,28)(23,38)(29,40)(31,33)(32,36)(39,43)(45,48)(46,47);(2,5)(3,7)(9,11,13,12)(10,14,15,16)(17,18,22,26)(19,29,28,40)(20,24,30,27)(21,35,37,41)(23,31,38,33)(25,42,44,34)(32,43,36,39)(45,46,48,47);(9,12,13,11)(10,14,15,16)(17,19,30,31,22,28,20,33)(18,23,24,40,26,38,27,29)(21,36,42,47,37,32,34,46)(25,43,41,45,44,39,35,48);(2,5)(6,8)(17,20,22,30)(18,24,26,27)(19,31,28,33)(21,34,37,42)(23,29,38,40)(25,41,44,35)(32,46,36,47)(39,45,43,48);(1,3,4,7)(2,6,5,8)(17,21,20,34,22,37,30,42)(18,25,27,35,26,44,24,41)(19,32,33,47,28,36,31,46)(23,39,29,45,38,43,40,48)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6727911);