#9065336

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

4494 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,e^d=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, 8972532);