Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| 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 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2210 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(14,20)(15,25)(16,21)(17,19)(22,27)(26,28);(2,4)(6,9)(10,12)(13,14)(15,26)(16,27)(17,23)(18,20)(19,24)(21,25)(22,28);(5,7,8,11)(6,10,9,12)(13,15,17,25,18,27,19,22)(14,21,24,26,20,28,23,16);(1,2,3,4)(13,16,19,28,18,26,17,21)(14,22,23,27,20,25,24,15);(6,9)(10,12)(13,17,18,19)(14,23,20,24)(15,22,27,25)(16,21,26,28)Polycyclic presentation:
2^9|c^2=fgi,d^2=h,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,e^c=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72280286, #72280333, #72280757, … |
| maximal normal subgroup of | #72280286, #72280333, #72280757, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7520870);