#6916326

order	512
nilpotency class	3
isoclinism	isoclinic class of #6885065
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	#25131	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1409 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)(11,15,16,14)(17,18)(19,24)(20,31)(21,26)(22,36)(23,27)(25,34)(28,45)(29,39)(30,40)(32,48)(33,44)(35,41)(37,43)(38,47)(42,46);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,23,29)(18,24,27,39)(20,32,33,28)(21,35,22,30)(25,42,43,38)(26,41,36,40)(31,48,44,45)(34,46,37,47);(3,7)(6,8)(10,13)(14,15)(17,20,23,33)(18,25,27,43)(19,28,29,32)(21,34,22,37)(24,38,39,42)(26,44,36,31)(30,46,35,47)(40,45,41,48);(1,3,4,7)(2,6,5,8)(9,11,12,16)(10,14,13,15)(19,29)(20,33)(24,39)(25,43)(30,35)(31,44)(34,37)(40,41);(2,5)(6,8)(17,21,23,22)(18,26,27,36)(19,30,29,35)(20,34,33,37)(24,40,39,41)(25,44,43,31)(28,46,32,47)(38,45,42,48)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6823522);