Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212×C4 |
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 | |
8670 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)(6,9)(7,11)(13,14)(15,20)(16,19)(17,21)(18,23)(22,26)(24,27)(25,28);(1,2)(3,4)(6,9)(10,12);(5,6,8,9)(7,10,11,12)(13,15)(14,18)(16,22)(17,23)(19,25)(20,21)(24,28)(26,27);(5,7,8,11)(6,10,9,12)(13,16,17,24)(14,19,21,27)(15,22,23,28)(18,25,20,26);(13,15)(14,20)(16,22)(17,23)(18,21)(19,26)(24,28)(25,27);(13,14)(15,18)(16,19)(17,21)(20,23)(22,25)(24,27)(26,28)Polycyclic presentation:
2^9|c^2=i,d^2=h,b^a=bg,c^a=ch,d^a=di,c^b=ci,f^e=fhi98 conjugacy classes — view the full character table.
| maximal subgroup of | #72283430, #72283529, #72286200, … |
| maximal normal subgroup of | #72283430, #72283529, #72286200, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7564138);