#404409

order	512
nilpotency class	3
isoclinism	isoclinic class of #394202
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:

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

Polycyclic presentation:

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

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, 311605);