#9070546

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

4331 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=gh,d^2=gh,e^2=gh,f^2=gh,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, 8977742);