#9137478

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

4400 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)(11,19)(12,17)(13,21)(14,15)(16,20)(18,23)(22,24)(25,26,31,28)(27,32,35,36)(29,33,39,38)(30,37,40,34);(1,2)(3,4)(5,7)(6,8)(10,15)(12,22)(16,19)(18,21)(25,27)(26,32)(28,36)(29,34)(30,33)(31,35)(37,39)(38,40);(1,3,5,8)(2,6,7,4)(10,15)(11,20)(17,24)(18,21)(26,28)(27,35)(33,38)(34,37);(3,8)(4,6)(9,11)(10,16)(12,18)(13,17)(14,20)(15,19)(21,22)(23,24)(25,28,31,26)(27,36,35,32)(29,33,39,38)(30,37,40,34);(1,4)(2,3)(5,6)(7,8)(9,12,14,22)(10,17,15,24)(11,21,20,18)(13,16,23,19)(25,29)(26,33)(27,37)(28,38)(30,36)(31,39)(32,40)(34,35);(1,2)(3,6)(4,8)(5,7)(9,13,14,23)(10,18,15,21)(11,17,20,24)(12,19,22,16)(25,30)(26,34)(27,33)(28,37)(29,32)(31,40)(35,38)(36,39)

Polycyclic presentation:

2^9|a^2=gh,c^2=ghi,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=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, 9044674);