#468915

order	512
nilpotency class	3
isoclinism	isoclinic class of #454025
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

886 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=egi,c^2=g,d^2=e,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, 376111);