#482146

order	512
nilpotency class	4
isoclinism	isoclinic class of #481619
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄⋊C₄	C₂⁴
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#519 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

856 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=gi,c^2=i,d^2=i,e^2=hi,f^2=i,b^a=be,c^a=cgh,e^a=eg,g^a=gi,e^b=eh,f^b=fghi,d^c=di,f^c=fi,f^e=fi

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, 389342);