Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4550 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(11,18)(12,15)(19,23)(20,21)(25,26,31,35)(27,36,39,32)(28,33,40,30)(29,37,38,34);(1,2)(3,7)(4,8)(5,6)(9,10)(11,12)(13,16)(14,17)(15,18)(19,20)(21,23)(22,24)(25,27)(26,32)(28,37)(29,33)(30,38)(31,39)(34,40)(35,36);(9,11,14,18)(10,15,17,12)(13,19,22,23)(16,21,24,20)(25,28,31,40)(26,30,35,33)(27,37,39,34)(29,32,38,36);(1,3,5,4)(2,7,6,8)(9,11)(10,12)(13,19)(14,18)(15,17)(16,20)(21,24)(22,23)(25,28)(26,33)(27,37)(29,32)(30,35)(31,40)(34,39)(36,38);(9,12)(10,11)(13,21)(14,15)(16,23)(17,18)(19,24)(20,22)(25,29,31,38)(26,34,35,37)(27,33,39,30)(28,32,40,36);(1,4,5,3)(2,8,6,7)(9,13,14,22)(10,16,17,24)(11,19,18,23)(12,20,15,21)(25,30,31,33)(26,28,35,40)(27,38,39,29)(32,37,36,34)Polycyclic presentation:
2^9|a^2=hi,c^2=h,d^2=ghi,e^2=hi,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7825868);