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 | |
7146 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)(5,6)(7,12)(8,10)(9,11)(14,19)(15,23)(21,22)(24,25);(1,2)(3,4)(14,19)(20,26)(21,22)(27,28);(5,7,9,10)(6,8,11,12)(13,14)(15,20)(16,21)(17,22)(18,19)(23,26)(24,27)(25,28);(13,15)(14,20)(16,24)(17,25)(18,23)(19,26)(21,27)(22,28);(5,6)(7,8)(9,11)(10,12)(13,16,18,17)(14,21,19,22)(15,24,23,25)(20,27,26,28);(5,8)(6,10)(7,11)(9,12)(13,17,18,16)(14,22,19,21)(15,25,23,24)(20,28,26,27)Polycyclic presentation:
2^9|c^2=hi,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=di,c^b=ci,f^e=fhi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7563946);