#6828692

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	#42906	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1451 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)(11,15)(12,14)(13,16)(17,18,22,26)(19,29,31,24)(20,33,32,25)(21,27,36,41)(23,28,34,39)(30,43,44,38)(35,46,45,40)(37,42,47,48);(9,11)(10,13)(12,16)(14,15)(17,19,22,31)(18,24,26,29)(20,34,32,23)(21,35,36,45)(25,39,33,28)(27,40,41,46)(30,37,44,47)(38,42,43,48);(2,5)(6,8)(10,14)(11,16)(17,20,23,19,22,32,34,31)(18,25,28,24,26,33,39,29)(21,35,37,30,36,45,47,44)(27,40,42,38,41,46,48,43);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(18,26)(21,36)(24,29)(25,33)(28,39)(30,44)(35,45)(37,47);(17,21,23,37,22,36,34,47)(18,27,28,42,26,41,39,48)(19,30,32,45,31,44,20,35)(24,38,33,46,29,43,25,40)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fhi,d^2=fg,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,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, 6735888);