#8587192

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

4498 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,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, 8494388);