#6506617

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500456
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

1632 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6413813);