#9028832

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

4493 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)(11,17)(12,20)(13,14)(15,18)(16,21)(19,22)(23,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,40)(37,38);(1,2,5,4)(3,7,8,6)(10,14)(15,17)(16,20)(22,23)(25,27,31,28)(26,32,30,33)(29,36,37,35)(34,40,38,39);(1,3,5,8)(2,6,4,7)(10,14)(11,18)(12,21)(22,23)(26,30)(27,28)(29,37)(39,40);(3,8)(6,7)(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,23)(21,24)(25,28,31,27)(26,33,30,32)(29,35,37,36)(34,39,38,40);(1,3)(2,7)(4,6)(5,8)(9,10)(11,17)(12,16)(13,14)(15,18)(19,23)(20,21)(22,24)(25,29,31,37)(26,34,30,38)(27,35,28,36)(32,39,33,40);(1,4,5,2)(3,6,8,7)(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,23)(18,24)(25,30,31,26)(27,33,28,32)(29,38,37,34)(35,40,36,39)

Polycyclic presentation:

2^9|b^2=i,c^2=ghi,d^2=gh,e^2=gh,f^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=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, 8936028);