#7014405

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1270 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)(17,18,22,39,23,29,24,28)(19,30,32,42,33,25,34,46)(20,27,38,26,21,37,36,44)(31,41,48,43,35,45,47,40);(1,3,5,8)(2,6,4,7)(9,10)(11,15)(12,14)(13,16)(17,19)(18,25)(20,35)(21,31)(22,32)(23,33)(24,34)(26,43)(27,41)(28,42)(29,30)(36,48)(37,45)(38,47)(39,46)(40,44);(1,2,5,4)(3,6,8,7)(9,11)(10,13)(12,16)(14,15)(17,20)(18,26)(19,31)(21,23)(22,36)(24,38)(25,40)(27,39)(28,37)(29,44)(30,43)(32,47)(33,35)(34,48)(41,42)(45,46);(3,8)(6,7)(11,16)(13,15)(17,21,23,20)(18,27,29,37)(19,31,33,35)(22,36,24,38)(25,41,30,45)(26,28,44,39)(32,48,34,47)(40,46,43,42);(1,4,5,2)(3,7,8,6)(10,14)(13,15)(17,22,23,24)(18,28,29,39)(19,32,33,34)(20,36,21,38)(25,42,30,46)(26,37,44,27)(31,47,35,48)(40,41,43,45)

Polycyclic presentation:

2^9|a^2=fg,b^2=fi,c^2=fi,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=di,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, 6921601);