Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1393169 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×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 | |
4189 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(6,7)(13,14)(15,20)(16,22)(17,21)(18,19)(23,25)(24,28)(26,27);(10,12)(14,19)(15,22)(21,27)(23,28);(1,2,4,3)(5,6,8,7)(9,10)(11,12)(13,15,18,22)(14,20,19,16)(17,23,26,28)(21,25,27,24);(5,6,8,7)(13,16,18,20)(14,15,19,22)(17,25,26,24)(21,28,27,23);(1,3,4,2)(5,7,8,6)(13,17)(14,21)(15,23)(16,24)(18,26)(19,27)(20,25)(22,28)Polycyclic presentation:
2^9|c^2=fg,d^2=i,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^c=df,e^d=ef86 conjugacy classes — view the full character table.
| maximal subgroup of | #72280497, #72280790, #72280830, … |
| maximal normal subgroup of | #72280497, #72280790, #72280830, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1316201);