#7010051

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1336 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fi,c^2=fi,d^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dfi,e^b=ehi,d^c=dh,g^c=gi,e^d=ei

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, 6917247);