#464248

order	512
nilpotency class	3
isoclinism	isoclinic class of #454426
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×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

1706 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)(9,10,15,21)(11,23,27,36)(12,29,32,18)(13,19,33,31)(14,22,16,20)(17,28,35,26)(24,38,34,37)(25,30,40,39);(5,6,7,8)(9,11,16,28,15,27,14,26)(10,17,22,36,21,35,20,23)(12,30,19,34,32,39,31,24)(13,25,18,37,33,40,29,38);(2,4)(6,8)(9,12)(10,18)(11,24)(13,22)(14,19)(15,32)(16,31)(17,25)(20,33)(21,29)(23,37)(26,30)(27,34)(28,39)(35,40)(36,38);(9,13,14,29,15,33,16,18)(10,19,20,12,21,31,22,32)(11,25,26,38,27,40,28,37)(17,34,23,30,35,24,36,39)

Polycyclic presentation:

2^9|a^2=i,b^2=g,d^2=e,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,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, 371444);