#8581485

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

4344 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8488681);