#467136

order	512
nilpotency class	3
isoclinism	isoclinic class of #454982
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1074 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 374332);