#402439

order	512
nilpotency class	3
isoclinism	isoclinic class of #394289
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1005 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=ei,c^2=efi,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 309635);