#482136

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×C₂²⋊C₄	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#520 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

911 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=i,e^2=hi,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, 389332);