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 | |
7109 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 28 points:
(2,4)(5,6,9,10)(7,11,12,8)(13,14,19,21)(15,20,25,28)(16,22,26,27)(17,23,24,18);(1,2)(3,4)(13,15,19,25)(14,20,21,28)(16,24,26,17)(18,27,23,22);(6,10)(7,12)(14,21)(15,25)(16,26)(18,23);(5,7)(6,11)(8,10)(9,12)(13,16)(14,22)(15,24)(17,25)(18,20)(19,26)(21,27)(23,28);(13,17,19,24)(14,23,21,18)(15,26,25,16)(20,27,28,22);(5,8,9,11)(6,7,10,12)(13,18,19,23)(14,24,21,17)(15,27,25,22)(16,20,26,28)Polycyclic presentation:
2^9|a^2=h,b^2=i,e^2=i,f^2=h,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,f^c=fi,e^d=ei110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7590737);