#467137

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	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1122 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)(9,10,13,18)(11,15)(12,16,23,33)(14,17,25,19)(20,31)(21,29)(22,28)(24,34,26,32)(27,35)(30,37)(36,39)(38,40);(1,3,4,7)(2,5,6,8)(9,11,13,21)(10,15,18,29)(12,20,23,35)(14,22,25,37)(16,31,33,27)(17,28,19,30)(24,36,26,38)(32,40,34,39);(3,7)(5,8)(9,12,13,23)(10,16,18,33)(11,20,21,35)(14,26,25,24)(15,27,29,31)(17,32,19,34)(22,38,37,36)(28,39,30,40);(10,17,18,19)(12,23)(15,28,29,30)(16,32,33,34)(20,35)(24,26)(27,39,31,40)(36,38)

Polycyclic presentation:

2^9|a^2=efg,b^2=eg,c^2=i,d^2=ei,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, 374333);