#6458043

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438955
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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1386 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)(3,6)(4,5)(7,8)(9,10,12,14)(11,13,15,16)(17,18)(19,24)(20,31)(21,26)(22,36)(23,27)(25,34)(28,42)(29,38)(30,39)(32,41)(33,45)(35,40)(37,43)(44,47)(46,48);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,28)(21,35)(22,29)(23,30)(25,37)(26,40)(27,39)(31,42)(32,46)(33,44)(34,43)(36,38)(41,48)(45,47);(11,15)(13,16)(17,20,22,34,23,33,21,32)(18,25,26,31,27,41,36,45)(19,28,29,43,30,44,35,46)(24,37,40,42,39,48,38,47);(11,15)(13,16)(17,21,23,22)(18,26,27,36)(19,29,30,35)(20,32,33,34)(24,38,39,40)(25,31,41,45)(28,43,44,46)(37,47,48,42);(1,3,4,7)(2,6,5,8)(19,30)(20,33)(24,39)(25,41)(29,35)(31,45)(32,34)(38,40)

Polycyclic presentation:

2^9|a^2=h,c^2=g,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dhi,e^b=ei,e^c=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6365239);