#409399

order	512
nilpotency class	3
isoclinism	isoclinic class of #401626
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#397
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1000 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

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, 316595);