#6812710

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806623
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	#42664	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1313 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6719906);