#419214

order	512
nilpotency class	3
isoclinism	isoclinic class of #417778
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1857 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)(11,21)(13,14)(17,32)(18,31)(19,20)(23,37)(24,25)(26,34)(27,28)(35,38)(36,39);(5,6)(7,8)(9,10,13,16,15,20,14,19)(11,22,24,32,25,30,21,17)(12,26,27,34,29,31,28,18)(23,33,35,39,38,40,37,36);(2,4)(6,8)(9,11)(10,17)(12,23)(13,21)(14,24)(15,25)(16,30)(18,33)(19,22)(20,32)(26,36)(27,37)(28,35)(29,38)(31,39)(34,40);(9,12,15,29)(10,18,20,34)(11,23,25,38)(13,28,14,27)(16,31,19,26)(17,33,32,40)(21,35,24,37)(22,36,30,39)

Polycyclic presentation:

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

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, 326410);