#475766

order	512
nilpotency class	3
isoclinism	isoclinic class of #457092
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23426	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1444 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 382962);