#9913330

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

4284 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)(4,6)(5,7)(9,10)(11,20)(12,22)(13,17)(14,21)(15,16)(18,23)(19,24)(27,33)(30,37)(34,39)(38,40);(1,3,5,6)(2,4,7,8)(10,16)(11,13)(18,22)(19,21)(25,26,29,32)(27,30,33,37)(28,31,36,35)(34,38,39,40);(9,11,15,13)(10,17,16,20)(12,19,23,21)(14,18,24,22)(25,26)(27,30)(28,31)(29,32)(33,37)(34,38)(35,36)(39,40);(9,12,15,23)(10,18,16,22)(11,21,13,19)(14,20,24,17)(25,27)(26,30)(28,34)(29,33)(31,38)(32,37)(35,40)(36,39);(1,4,5,8)(2,6,7,3)(9,13,15,11)(10,17,16,20)(12,19,23,21)(14,22,24,18)(25,26)(27,30)(28,35)(29,32)(31,36)(33,37)(34,40)(38,39);(1,2)(3,4)(5,7)(6,8)(9,14)(10,19)(11,18)(12,20)(13,22)(15,24)(16,21)(17,23)(25,28)(26,31)(27,34)(29,36)(30,38)(32,35)(33,39)(37,40)

Polycyclic presentation:

2^9|b^2=gi,c^2=h,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9820526);