Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514196 |
| 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 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2446 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(5,6)(7,11)(8,12)(9,10)(13,14)(15,19)(16,20)(17,22)(18,21)(23,26)(24,27)(25,28);(2,3)(6,10)(11,12)(13,15,18,25)(14,19,21,28)(16,23,17,24)(20,26,22,27);(1,2,4,3)(5,7,9,8)(6,11,10,12)(13,16,18,17)(14,20,21,22)(15,23,25,24)(19,26,28,27);(1,3,4,2)(5,8,9,7)(6,12,10,11)(15,24)(16,17)(19,27)(20,22)(23,25)(26,28);(6,10)(11,12)(14,21)(19,28)(20,22)(26,27)Polycyclic presentation:
2^9|b^2=i,c^2=fgi,d^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,d^c=di,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72283039, #72283663, #72283909, … |
| maximal normal subgroup of | #72283039, #72283663, #72283909, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7468847);