#393531

order	512
nilpotency class	3
isoclinism	isoclinic class of #196248
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1657 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)(3,6)(4,5)(7,8)(9,10,14,19,15,18,13,28)(11,20,24,30,23,34,22,16)(12,17,26,32,27,31,25,38)(21,33,37,39,36,40,35,29);(9,11)(10,16,18,30)(12,21)(13,22)(14,24)(15,23)(17,29,31,39)(19,20,28,34)(25,35)(26,37)(27,36)(32,33,38,40);(2,5)(6,8)(9,12,15,27)(10,17,18,31)(11,21,23,36)(13,25,14,26)(16,29,30,39)(19,32,28,38)(20,33,34,40)(22,35,24,37);(1,3,4,7)(2,6,5,8)(10,18)(13,14)(16,30)(17,31)(22,24)(25,26)(29,39)(35,37)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 300727);