Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #635984 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C211 |
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 | |
7698 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)(14,18)(15,21)(16,24)(26,28);(10,12)(13,14,17,18)(15,19,21,22)(16,20,24,25)(23,26,27,28);(5,6)(7,8)(9,10)(11,12)(13,14)(15,22)(16,20)(17,18)(19,21)(23,28)(24,25)(26,27);(1,2)(3,4)(5,6)(7,8)(13,15)(14,19)(16,23)(17,21)(18,22)(20,26)(24,27)(25,28);(13,16)(14,20)(15,23)(17,24)(18,25)(19,26)(21,27)(22,28)Polycyclic presentation:
2^9|b^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^c=df92 conjugacy classes — view the full character table.
| maximal subgroup of | #72304737, #72304741, #72304746, … |
| maximal normal subgroup of | #72304737, #72304741, #72304746, … |
| maximal quotient of | #72304982, #72304986, #72305037, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 543181);