#8551131

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

4671 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,f^2=hi,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, 8458327);