#464192

order	512
nilpotency class	3
isoclinism	isoclinic class of #457092
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	#23422	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1793 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)(5,6)(7,8)(9,10)(11,21)(12,26)(13,19)(14,20)(15,18)(16,24)(17,29)(22,31)(23,30)(25,32)(27,34)(28,33)(35,37)(36,40)(38,39);(9,11)(10,16,20,32)(12,23)(13,24)(14,22)(15,25)(17,30,34,40)(18,21,19,31)(26,39,33,35)(27,38)(28,36)(29,37);(2,4)(11,22)(12,27)(13,15)(17,33)(18,19)(21,31)(23,37)(26,34)(28,29)(30,35)(36,38)(39,40);(6,8)(9,12,15,29,14,28,13,27)(10,17,19,26,20,34,18,33)(11,23,25,37,22,36,24,38)(16,30,31,39,32,40,21,35)

Polycyclic presentation:

2^9|b^2=gi,d^2=ei,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^c=de,e^c=ei,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, 371388);