#391380

order	512
nilpotency class	3
isoclinism	isoclinic class of #219525
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

773 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 298576);