#9090052

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

4376 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=gh,d^2=g,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,e^c=egi,e^d=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8997248);