#9125691

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

4541 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,d^2=g,e^2=gh,f^2=gh,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, 9032887);