#6912696

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907909
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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1614 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gi,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,d^c=dfh

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, 6819892);