Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #616960 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
7414 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(6,8)(14,15)(16,23)(17,25)(26,27);(10,12)(13,14,18,15)(16,21,23,19)(17,20,25,22)(24,27,28,26);(5,6)(7,8)(9,10,11,12)(13,15,18,14)(16,21,23,19)(17,22,25,20)(24,27,28,26);(1,2)(3,4)(5,6)(7,8)(13,16,18,23)(14,19,15,21)(17,24,25,28)(20,26,22,27);(13,17)(14,20)(15,22)(16,24)(18,25)(19,26)(21,27)(23,28)Polycyclic presentation:
2^9|b^2=f,c^2=fh,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=df92 conjugacy classes — view the full character table.
| maximal subgroup of | #72315813 |
| maximal normal subgroup of | #72315813 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 524909);