#9909799

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

4671 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)(3,7)(4,6)(5,8)(9,10,14,16)(11,15,19,23)(12,17,22,21)(13,20,24,18)(25,26)(27,36)(28,38)(29,33)(30,37)(31,32)(34,39)(35,40);(1,2,4,6)(3,5,8,7)(9,11)(10,15)(12,20)(13,21)(14,19)(16,23)(17,24)(18,22)(26,32)(27,29)(34,38)(35,37);(1,2)(3,5)(4,6)(7,8)(10,16)(11,19)(17,21)(18,20)(25,27,31,29)(26,33,32,36)(28,35,39,37)(30,34,40,38);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,20)(13,21)(14,19)(16,23)(17,24)(18,22)(25,28,31,39)(26,34,32,38)(27,37,29,35)(30,36,40,33);(9,12)(10,17)(11,20)(13,23)(14,22)(15,24)(16,21)(18,19)(25,29,31,27)(26,33,32,36)(28,35,39,37)(30,38,40,34);(9,13,14,24)(10,18,16,20)(11,21,19,17)(12,15,22,23)(25,30)(26,35)(27,34)(28,36)(29,38)(31,40)(32,37)(33,39)

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=ghi,d^2=ghi,e^2=ghi,f^2=gi,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, 9816995);