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 | |
8652 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(2,4)(6,10)(7,11)(13,14)(15,20)(16,18)(17,22)(19,21)(23,28)(24,25)(26,27);(1,2)(3,4)(5,6,9,10)(7,8,11,12)(13,15)(14,20)(16,24)(17,19)(18,25)(21,22)(23,26)(27,28);(5,7)(6,8)(9,11)(10,12)(14,21)(15,17)(16,23)(25,27);(5,8)(6,11)(7,10)(9,12)(13,16,19,23)(14,18,21,28)(15,24,17,26)(20,25,22,27);(13,17,19,15)(14,22,21,20)(16,26,23,24)(18,27,28,25);(13,18)(14,23)(15,25)(16,21)(17,27)(19,28)(20,26)(22,24)Polycyclic presentation:
2^9|b^2=hi,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, 7586362);