#10159181

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

4323 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)(11,13)(12,16)(14,18)(15,22)(17,23)(19,24)(20,21)(26,32)(28,37)(33,34)(35,38);(2,6)(3,7)(9,11,14,17)(10,13,18,23)(12,20,22,19)(15,24,16,21)(25,26)(27,33)(28,30)(29,34)(31,32)(35,36)(37,40)(38,39);(1,3)(2,4)(5,7)(6,8)(9,12,14,22)(10,15,18,16)(11,19,17,20)(13,21,23,24)(25,27,31,29)(26,33,32,34)(28,35,37,38)(30,36,40,39);(1,4)(2,3)(5,8)(6,7)(9,12)(10,16)(11,20)(13,21)(14,22)(15,18)(17,19)(23,24)(25,28)(26,30)(27,35)(29,38)(31,37)(32,40)(33,36)(34,39);(9,11)(10,13)(12,19)(14,17)(15,21)(16,24)(18,23)(20,22)(25,29,31,27)(26,34,32,33)(28,35,37,38)(30,36,40,39);(9,13)(10,17)(11,18)(12,21)(14,23)(15,20)(16,19)(22,24)(25,30)(26,28)(27,36)(29,39)(31,40)(32,37)(33,35)(34,38)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gi,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10066377);