#464244

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	#23426	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1754 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)(10,16)(12,26)(13,14)(17,21)(18,34)(19,20)(23,24)(27,38)(30,36)(31,37)(32,33)(35,40);(5,6,7,8)(9,10,14,19,15,20,13,16)(11,21,24,33,25,32,23,17)(12,18,29,34,26,36,28,30)(22,37,38,40,39,35,27,31);(2,4)(6,8)(9,11)(10,17)(12,27)(13,24)(14,23)(15,25)(16,21)(18,35)(19,32)(20,33)(22,28)(26,38)(29,39)(30,31)(34,40)(36,37);(9,12,13,28,15,26,14,29)(10,18,16,30,20,36,19,34)(11,22,23,27,25,39,24,38)(17,31,32,35,33,40,21,37)

Polycyclic presentation:

2^9|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, 371440);