#6999261

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

1377 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)(17,18,23,29)(19,31,35,25)(20,36,21,27)(22,30,24,28)(26,38,44,37)(32,41,33,45)(34,42,39,46)(40,48,43,47);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,14)(13,16)(17,19,23,35)(18,25,29,31)(20,33,21,32)(22,39,24,34)(26,43,44,40)(27,45,36,41)(28,46,30,42)(37,47,38,48);(9,11)(10,13)(12,16)(14,15)(17,20,23,21)(18,26,29,44)(19,32,35,33)(22,37,24,38)(25,40,31,43)(27,28,36,30)(34,47,39,48)(41,46,45,42);(3,8)(6,7)(11,16)(13,15)(17,21,23,20)(18,27,29,36)(19,33,35,32)(22,37,24,38)(25,41,31,45)(26,28,44,30)(34,47,39,48)(40,46,43,42);(1,4,5,2)(3,7,8,6)(10,14)(13,15)(17,22,23,24)(18,28,29,30)(19,34,35,39)(20,37,21,38)(25,42,31,46)(26,36,44,27)(32,47,33,48)(40,41,43,45)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,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, 6906457);