#9224333

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

4615 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=h,d^2=i,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,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, 9131529);