#7090733

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996355
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	C₂×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1595 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,21,27,22,26,33,37)(19,24,30,39,29,41,36,23)(20,25,34,44,35,43,45,48)(28,40,31,46,32,42,47,38);(9,10)(11,15)(12,14)(13,16)(17,19,22,29)(18,23,26,39)(20,31,35,47)(21,36,33,30)(24,27,41,37)(25,40,43,42)(28,45,32,34)(38,48,46,44);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,19)(18,24)(20,32)(21,30)(22,29)(23,37)(25,42)(26,41)(27,39)(28,35)(31,45)(33,36)(34,47)(38,44)(40,43)(46,48);(11,16)(13,15)(17,20,33,45,22,35,21,34)(18,25,37,48,26,43,27,44)(19,28,36,47,29,32,30,31)(23,38,41,42,39,46,24,40);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(18,26)(19,29)(25,43)(27,37)(28,32)(30,36)(31,47)(44,48)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=ghi,d^2=fi,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=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, 6997929);