#273546

order	512
nilpotency class	3
isoclinism	isoclinic class of #273435
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1083 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)(3,5)(4,7)(6,8)(9,10)(11,19,23,16)(12,14)(13,24,27,17)(15,18)(20,22,32,29)(21,30)(25,28)(26,37,39,31)(33,35)(34,36)(38,40);(2,5)(3,8)(6,7)(9,11)(10,16)(12,19)(13,21)(14,23)(15,29)(17,30)(18,32)(20,25)(22,28)(24,33)(26,38)(27,35)(31,40)(34,37)(36,39);(9,12,14,10)(11,20,23,32)(13,24,27,17)(15,18,28,25)(16,29,19,22)(21,34,35,40)(26,31,39,37)(30,38,33,36);(2,5)(3,8)(6,7)(9,13,14,27)(10,17,12,24)(11,21,23,35)(15,26,28,39)(16,30,19,33)(18,31,25,37)(20,34,32,40)(22,36,29,38)

Polycyclic presentation:

2^9|a^2=fhi,c^2=hi,d^2=hi,e^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,g^b=ghi,e^c=ehi,g^c=ghi,e^d=ei,f^d=fhi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 180742);