#475768

order	512
nilpotency class	3
isoclinism	isoclinic class of #457096
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	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1386 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)(9,10,15,20)(11,21,23,32)(12,17,29,36)(13,19,14,18)(16,25,31,24)(22,33,38,39)(26,30,37,40)(27,35,28,34);(5,6,7,8)(9,11,14,25,15,23,13,24)(10,16,19,32,20,31,18,21)(12,26,27,38,29,37,28,22)(17,33,34,40,36,39,35,30);(2,4)(6,8)(9,12)(10,17)(11,22)(13,27)(14,28)(15,29)(16,30)(18,34)(19,35)(20,36)(21,33)(23,38)(24,26)(25,37)(31,40)(32,39);(11,23)(12,27,29,28)(16,31)(17,34,36,35)(21,32)(22,37,38,26)(24,25)(30,39,40,33)

Polycyclic presentation:

2^9|a^2=i,b^2=g,d^2=h,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^b=di,d^c=de,e^c=ei,g^c=gi,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, 382964);