#393533

order	512
nilpotency class	3
isoclinism	isoclinic class of #196320
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	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1709 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)(11,15,21,28)(12,16)(13,24,25,17)(14,18)(19,26,32,38)(20,27)(22,34,35,29)(23,30)(31,37)(33,36)(39,40);(9,11)(10,15,18,28)(12,19)(13,20)(14,21)(16,26,30,38)(17,27,24,36)(22,31)(23,32)(25,33)(29,37,34,40)(35,39);(2,5)(6,8)(9,12,14,23)(10,16,18,30)(11,19,21,32)(13,22,25,35)(15,26,28,38)(17,29,24,34)(20,31,33,39)(27,37,36,40);(1,3,4,7)(2,6,5,8)(9,13)(10,17)(11,20)(12,22)(14,25)(15,27)(16,29)(18,24)(19,31)(21,33)(23,35)(26,37)(28,36)(30,34)(32,39)(38,40)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=i,d^2=f,b^a=be,c^a=cf,d^a=dg,e^a=ei,g^a=gi,e^b=ei,g^b=gi,h^b=hi,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, 300729);