Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #686200 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C22×#546 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
6228 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)(6,7)(13,14)(15,20)(16,17)(18,19)(21,26)(22,28)(23,25)(24,27);(10,12)(14,19)(17,21)(20,27)(23,28);(5,6,8,7)(9,10,11,12)(13,15)(14,20)(16,25)(17,23)(18,24)(19,27)(21,28)(22,26);(1,2,3,4)(5,7,8,6)(13,16)(14,21)(15,22)(17,19)(18,26)(20,23)(24,25)(27,28);(13,17,18,21)(14,16,19,26)(15,23,24,28)(20,25,27,22)Polycyclic presentation:
2^9|c^2=gh,d^2=fi,e^2=f,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=df92 conjugacy classes — view the full character table.
| maximal subgroup of | #72280804, #72280912 |
| maximal normal subgroup of | #72280804, #72280912 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 626895);