Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3858029 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2971 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)(7,11)(8,9)(10,12)(14,17)(15,19)(21,22,26,28)(23,25,24,27);(2,4)(6,9)(7,12)(13,14,16,17)(15,20,19,18)(21,23,26,24)(22,27,28,25);(1,2,3,4)(7,12)(10,11)(14,17)(18,20)(21,23,26,24)(22,27,28,25);(5,6)(7,11)(8,9)(10,12)(13,15)(14,18)(16,19)(17,20)(21,24,26,23)(22,25,28,27);(5,7)(6,10)(8,12)(9,11)(13,14)(15,20)(16,17)(18,19)(21,25)(22,24)(23,28)(26,27)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=hi,d^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,e^b=ef,e^c=ef,e^d=ef86 conjugacy classes — view the full character table.
| maximal subgroup of | #72336534, #72336545, #72336590 |
| maximal normal subgroup of | #72336534, #72336545, #72336590 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3851381);