#9726533

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

4446 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)(6,7)(9,10)(11,13)(12,19)(14,22)(15,16)(17,24)(18,23)(20,21)(25,26)(27,33)(28,32)(29,34)(30,31)(35,40)(36,37)(38,39);(1,2,5,4)(3,7,8,6)(9,11,15,12)(10,13,16,19)(14,18,24,20)(17,21,22,23)(25,27)(26,29)(28,36)(30,34)(31,33)(32,40)(35,38)(37,39);(1,3,5,8)(2,6,4,7)(10,16)(11,12)(17,22)(18,20)(26,31)(27,34)(32,39)(35,36);(1,4,5,2)(3,7,8,6)(9,12,15,11)(10,13,16,19)(14,20,24,18)(17,21,22,23)(25,27)(26,29)(28,35)(30,34)(31,33)(32,37)(36,38)(39,40);(9,13)(10,11)(12,16)(14,23)(15,19)(17,20)(18,22)(21,24)(25,28,30,38)(26,32,31,39)(27,35,34,36)(29,37,33,40);(9,14,15,24)(10,17,16,22)(11,18,12,20)(13,21,19,23)(25,29)(26,27)(28,37)(30,33)(31,34)(32,35)(36,39)(38,40)

Polycyclic presentation:

2^9|b^2=gh,c^2=hi,d^2=gh,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=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, 9633729);