Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
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 | |
8190 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2)(3,4)(5,6)(7,11)(8,9)(10,12)(13,14)(15,20)(16,18)(17,22)(19,21)(23,27)(24,26)(25,28);(5,7)(6,10)(8,11)(9,12)(13,15,19,16)(14,20,21,18)(17,24,25,23)(22,26,28,27);(2,4)(5,8,9,6)(7,11,12,10)(14,21)(15,16)(17,25)(26,27);(1,2)(3,4)(5,7)(6,11)(8,10)(9,12)(13,16,19,15)(14,18,21,20)(17,24,25,23)(22,26,28,27);(13,17,19,25)(14,22,21,28)(15,23,16,24)(18,26,20,27);(13,18)(14,15)(16,21)(17,26)(19,20)(22,23)(24,28)(25,27)Polycyclic presentation:
2^9|b^2=i,c^2=g,d^2=i,e^2=i,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7586343);