#9909769

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

4710 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=ghi,d^2=ghi,e^2=ghi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,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, 9816965);