#6653219

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632074
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 ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1523 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)(3,5)(4,7)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18,23,29)(19,31,35,25)(20,27,21,38)(22,30,24,28)(26,39,44,37)(32,46,33,41)(34,42,36,43)(40,47,45,48);(1,3,5,2)(4,7,8,6)(9,11,12,16)(10,13,14,15)(17,19,23,35)(18,25,29,31)(20,32,21,33)(22,34,24,36)(26,40,44,45)(27,41,38,46)(28,42,30,43)(37,47,39,48);(11,16)(13,15)(17,20,23,21)(18,26,29,44)(19,32,35,33)(22,37,24,39)(25,40,31,45)(27,30,38,28)(34,47,36,48)(41,43,46,42);(2,3)(6,7)(10,14)(13,15)(17,21,23,20)(18,27,29,38)(19,33,35,32)(22,37,24,39)(25,41,31,46)(26,30,44,28)(34,47,36,48)(40,43,45,42);(1,4,5,8)(2,6,3,7)(17,22,23,24)(18,28,29,30)(19,34,35,36)(20,37,21,39)(25,42,31,43)(26,27,44,38)(32,47,33,48)(40,41,45,46)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,d^2=i,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=dfhi,g^c=gi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6560415);