#7080617

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1576 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)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,24,39,23,28,22,27)(19,29,21,25,31,45,32,42)(20,33,40,44,36,26,37,48)(30,47,38,43,34,46,35,41);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(17,19,23,31)(18,25,28,42)(20,34,36,30)(21,24,32,22)(26,43,33,41)(27,45,39,29)(35,40,38,37)(44,46,48,47);(2,6)(5,7)(9,11,12,16)(10,13,14,15)(17,20,23,36)(18,26,28,33)(19,30,31,34)(21,38,32,35)(22,37,24,40)(25,41,42,43)(27,44,39,48)(29,46,45,47);(3,8)(5,7)(11,16)(13,15)(17,21,22,19,23,32,24,31)(18,25,27,29,28,42,39,45)(20,35,37,34,36,38,40,30)(26,43,44,47,33,41,48,46);(10,14)(13,15)(19,31)(20,36)(21,32)(25,42)(26,33)(29,45)(37,40)(44,48)

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=h,d^2=fgi,f^2=i,b^a=bf,c^a=cg,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,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, 6987813);