#8471030

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4521 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=hi,d^2=hi,e^2=hi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8378226);