Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667580 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2897 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6)(7,11)(8,10)(9,12);(5,7)(6,9)(8,12)(10,11)(14,18)(15,21)(16,24)(19,25)(23,26)(27,28);(2,4)(13,14)(15,22)(16,25)(17,19)(18,24)(20,23)(21,28)(26,27);(6,10)(9,11)(14,19)(18,25)(20,22)(27,28);(7,12)(9,11)(13,15,17,23)(14,20,19,22)(16,26,24,21)(18,27,25,28)Polycyclic presentation:
2^9|e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72281826, #72281834, #72281966, … |
| maximal normal subgroup of | #72281826, #72281834, #72281966, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6574777);