Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514060 |
| 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 |
| prime radical | #36160 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2077 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)(8,12)(9,10)(13,14)(15,18)(16,20)(17,19)(21,28)(22,25)(23,26)(24,27);(1,2)(3,4)(6,10)(11,12)(13,15,17,23)(14,18,19,26)(16,21,24,22)(20,28,27,25);(2,4);(2,4)(5,7,9,8)(6,11,10,12)(14,19)(15,21)(16,24)(18,25)(22,23)(26,28);(5,8,9,7)(6,11,10,12)Polycyclic presentation:
2^9|b^2=i,d^2=f,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,h^b=hi,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72280126, #72308710, #72308876, … |
| maximal normal subgroup of | #72280126, #72308710, #72308876, … |
| maximal quotient of | #72311726 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7428101);