#10086345

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

4512 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)(4,5)(6,8)(9,10)(11,16)(12,15)(13,17)(14,22)(18,23)(19,20)(21,24)(25,26)(27,30)(28,35)(29,32)(31,36)(33,37)(34,40)(38,39);(2,5)(6,7)(9,11,15,19)(10,16,12,20)(13,23,24,14)(17,18,21,22)(27,33)(30,37)(34,39)(38,40);(1,2)(3,6)(4,5)(7,8)(10,12)(11,19)(13,24)(18,22)(25,26,29,32)(27,30,33,37)(28,31,36,35)(34,38,39,40);(1,3)(2,6)(4,8)(5,7)(9,12,15,10)(11,16,19,20)(13,21,24,17)(14,22,23,18)(25,26)(27,30)(28,31)(29,32)(33,37)(34,38)(35,36)(39,40);(9,13,15,24)(10,17,12,21)(11,14,19,23)(16,22,20,18)(25,27)(26,30)(28,34)(29,33)(31,38)(32,37)(35,40)(36,39);(9,14)(10,18)(11,13)(12,22)(15,23)(16,21)(17,20)(19,24)(25,28,29,36)(26,31,32,35)(27,34,33,39)(30,38,37,40)

Polycyclic presentation:

2^9|b^2=gi,c^2=ghi,d^2=gi,e^2=gi,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,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, 9993541);