#369363

order	512
nilpotency class	3
isoclinism	isoclinic class of #196152
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1412 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,c^2=efg,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,f^b=fi,g^b=gi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 276559);