#6833233

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1235 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)(9,10,12,13)(11,14,15,16)(17,18)(19,30)(20,33)(21,39)(22,28)(23,29)(24,27)(25,32)(26,37)(31,41)(34,43)(35,44)(36,45)(38,48)(40,42)(46,47);(1,2)(3,6)(4,5)(7,8)(17,19,23,32)(18,25,29,30)(20,34,37,36)(21,22,24,40)(26,43,33,45)(27,28,39,42)(31,35,47,38)(41,48,46,44);(10,13)(14,16)(17,20,21,35,23,37,24,38)(18,26,39,48,29,33,27,44)(19,31,40,36,32,47,22,34)(25,41,28,45,30,46,42,43);(1,3,4,7)(2,6,5,8)(17,21,23,24)(18,27,29,39)(19,22,32,40)(20,35,37,38)(25,28,30,42)(26,44,33,48)(31,34,47,36)(41,45,46,43);(2,5)(6,8)(9,11,12,15)(10,14,13,16)(17,22,21,19,23,40,24,32)(18,28,39,30,29,42,27,25)(20,36,35,47,37,34,38,31)(26,43,48,41,33,45,44,46)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=hi,d^2=f,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, 6740429);