#404403

order	512
nilpotency class	3
isoclinism	isoclinic class of #394292
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1141 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=i,d^2=g,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,f^c=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, 311599);