#6432662

order	512
nilpotency class	3
isoclinism	isoclinic class of #6431707
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1814 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339858);