#409524

order	512
nilpotency class	3
isoclinism	isoclinic class of #402446
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2504
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

751 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:

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

Polycyclic presentation:

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

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, 316720);