#6457124

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1758 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)(19,30)(20,36)(21,26)(22,27)(23,29)(24,34)(25,37)(28,31)(32,46)(33,43)(35,44)(38,45)(39,47)(40,48)(41,42);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19,31,33,23,35,22,34)(18,24,27,44,29,43,28,30)(20,21,32,39,38,41,37,40)(25,42,45,47,46,26,36,48);(1,3,4,7)(2,6,5,8)(11,15)(13,16)(17,20,22,37,23,38,31,32)(18,25,28,36,29,46,27,45)(19,21,34,40,35,41,33,39)(24,42,30,48,43,26,44,47);(11,15)(13,16)(17,21)(18,26)(19,32)(20,34)(22,40)(23,41)(24,36)(25,44)(27,48)(28,47)(29,42)(30,46)(31,39)(33,38)(35,37)(43,45);(17,22,23,31)(18,27,29,28)(19,33,35,34)(20,37,38,32)(21,39,41,40)(24,30,43,44)(25,45,46,36)(26,47,42,48)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=f,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^b=ei,e^d=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, 6364320);