#8232461

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

4454 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:

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

Polycyclic presentation:

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

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, 8139657);