#6834291

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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	#42992	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1639 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)(3,7)(4,6)(5,8)(9,10)(11,16)(12,14)(13,15)(17,18,23,28)(19,30,33,24)(20,34,38,25)(21,29,39,26)(22,27,32,41)(31,40,48,46)(35,42,37,45)(36,43,47,44);(1,3)(2,5)(4,8)(6,7)(17,19,23,33)(18,24,28,30)(20,35,38,37)(21,32,39,22)(25,42,34,45)(26,41,29,27)(31,47,48,36)(40,43,46,44);(9,11,13,12)(10,14,15,16)(17,20,21,36,23,38,39,47)(18,25,29,44,28,34,26,43)(19,31,22,35,33,48,32,37)(24,40,41,42,30,46,27,45);(2,6)(5,7)(17,21,23,39)(18,26,28,29)(19,32,33,22)(20,36,38,47)(24,41,30,27)(25,43,34,44)(31,37,48,35)(40,42,46,45);(3,8)(5,7)(9,12,13,11)(10,14,15,16)(17,22,39,19,23,32,21,33)(18,27,26,30,28,41,29,24)(20,37,47,48,38,35,36,31)(25,42,43,40,34,45,44,46)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=ghi,d^2=i,e^2=gh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,e^d=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, 6741487);