#418981

order	512
nilpotency class	3
isoclinism	isoclinic class of #417603
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×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

2224 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)(9,10,14,21,16,22,15,20)(11,23,25,17,26,28,18,12)(13,29,31,19,33,36,32,37)(24,34,30,38,40,27,39,35);(5,6)(7,8)(9,11,14,25,16,26,15,18)(10,17,20,23,22,12,21,28)(13,30,31,40,33,39,32,24)(19,35,29,27,37,38,36,34);(2,4)(6,8)(9,12,16,17)(10,18,22,25)(11,21,26,20)(13,27,33,34)(14,28,15,23)(19,30,37,39)(24,29,40,36)(31,38,32,35);(9,13,16,33)(10,19,22,37)(11,24,26,40)(12,27,17,34)(14,32,15,31)(18,30,25,39)(20,36,21,29)(23,38,28,35)

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=i,d^2=i,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,f^c=fi,h^c=hi,e^d=ei,f^d=fi,h^d=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, 326177);