#394375

order	512
nilpotency class	3
isoclinism	isoclinic class of #393925
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2555 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ei,d^2=i,e^2=i,g^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,e^c=ei,g^c=gi,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 301571);