#9125693

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

4382 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=g,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, 9032889);