#6834338

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	#2388	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1610 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)(19,29)(20,33)(21,36)(22,26)(23,28)(24,31)(25,32)(27,37)(30,39)(34,40)(35,41)(38,48)(42,47)(43,44)(45,46);(1,3)(2,5)(4,8)(6,7)(17,19,22,31)(18,24,26,29)(20,34,32,23)(21,35,37,45)(25,40,33,28)(27,41,36,46)(30,38,44,47)(39,42,43,48);(9,11,13,12)(10,14,15,16)(17,20,23,19,22,32,34,31)(18,25,28,24,26,33,40,29)(21,35,38,30,37,45,47,44)(27,41,42,39,36,46,48,43);(2,6)(5,7)(18,26)(21,37)(24,29)(25,33)(28,40)(30,44)(35,45)(38,47);(3,8)(5,7)(9,12,13,11)(10,14,15,16)(17,21,23,38,22,37,34,47)(18,27,28,42,26,36,40,48)(19,30,32,45,31,44,20,35)(24,39,33,46,29,43,25,41)

Polycyclic presentation:

2^9|b^2=i,c^2=ghi,e^2=ghi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,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, 6741534);