#419025

order	512
nilpotency class	3
isoclinism	isoclinic class of #417784
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
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2296 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:

(2,4)(9,10,14,21,16,22,15,20)(11,23,25,17,27,36,26,37)(12,28,30,39,24,38,29,18)(13,31,33,19,34,35,32,40);(1,2,3,4)(5,6)(7,8)(9,11,14,25,16,27,15,26)(10,17,20,23,22,37,21,36)(12,13,29,32,24,34,30,33)(18,35,28,40,39,31,38,19);(6,8)(9,12)(10,18)(11,13)(14,29)(15,30)(16,24)(17,35)(19,36)(20,28)(21,38)(22,39)(23,40)(25,32)(26,33)(27,34)(31,37);(2,4)(9,13,16,34)(10,19,22,40)(11,24,27,12)(14,33,15,32)(17,28,37,38)(18,36,39,23)(20,31,21,35)(25,29,26,30)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 326221);