Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499814 |
| 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 ◁ #2283 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2266 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(15,22)(16,17)(19,27)(20,26)(23,28)(24,25);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,23)(16,26)(17,20)(18,21)(19,25)(22,28)(24,27);(6,9)(7,11)(13,15)(14,19)(16,22)(17,24)(18,25)(20,28)(21,23)(26,27);(5,7,8,11)(6,10,9,12)(13,16,18,17)(14,20,21,26)(15,24,25,22)(19,27,23,28);(6,9)(10,12)(13,16,18,17)(14,20,21,26)(15,22,25,24)(19,28,23,27)Polycyclic presentation:
2^9|b^2=f,d^2=h,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72314184, #72314242, #72316228, … |
| maximal normal subgroup of | #72314184, #72314242, #72316228, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6420674);