Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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 | |
2860 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)(5,6)(7,11)(8,9)(10,12)(14,16)(15,19)(21,22,25,26)(23,24,27,28);(6,9)(10,11)(15,19)(18,20)(21,23,25,27)(22,24,26,28);(7,12)(10,11)(13,14,17,16)(15,18,19,20)(21,24)(22,23)(25,28)(26,27);(1,2,3,4)(5,7,8,12)(6,10,9,11)(13,15,17,19)(14,18,16,20)(21,23)(22,24)(25,27)(26,28);(5,6)(7,10)(8,9)(11,12)(13,16,17,14)(15,20,19,18)(21,22)(23,24)(25,26)(27,28)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=gh,d^2=i,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
| maximal subgroup of | #72327815, #72327855, #72327882 |
| maximal normal subgroup of | #72327815, #72327855, #72327882 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1555705);