#8358528

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

4333 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=g,d^2=gi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8265724);