#6432503

order	512
nilpotency class	3
isoclinism	isoclinic class of #6431710
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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1786 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)(11,13)(12,14)(15,16)(17,18,22,25)(19,23,28,38)(20,24,32,40)(21,26,35,42)(27,36,45,47)(29,37,33,43)(30,39,46,48)(31,41,34,44);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(17,19)(18,23)(20,30)(21,33)(22,28)(24,39)(25,38)(26,43)(27,31)(29,35)(32,46)(34,45)(36,41)(37,42)(40,48)(44,47);(10,14)(13,16)(17,20,22,32)(18,24,25,40)(19,27,28,45)(21,34,35,31)(23,36,38,47)(26,44,42,41)(29,46,33,30)(37,48,43,39);(1,3,4,7)(2,6,5,8)(17,19)(18,23)(20,27)(21,29)(22,28)(24,36)(25,38)(26,37)(30,31)(32,45)(33,35)(34,46)(39,41)(40,47)(42,43)(44,48);(18,25)(19,28)(24,40)(26,42)(27,45)(29,33)(30,46)(41,44)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339699);