#468822

order	512
nilpotency class	3
isoclinism	isoclinic class of #454025
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
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

1231 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2)(3,6)(4,5)(7,8)(11,15)(13,16)(17,18,21,26,20,23,22,31)(19,27,29,25,28,32,30,24);(2,5)(6,8)(9,10,12,14)(11,16,15,13)(18,23)(24,25)(26,31)(27,32);(1,3)(2,6)(4,7)(5,8)(9,11,12,15)(10,13,14,16)(17,19,20,28)(18,24,23,25)(21,30,22,29)(26,32,31,27);(3,7)(6,8)(17,19)(18,25)(20,28)(21,29)(22,30)(23,24)(26,32)(27,31)

Polycyclic presentation:

2^9|a^2=fg,b^2=g,c^2=gi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^c=de,f^c=fi,h^d=hi

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, 376018);