#409527

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

Polycyclic presentation:

2^9|a^2=egi,b^2=h,c^2=ef,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,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, 316723);