Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #988785 |
| automorphism | ? |
| 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 | |
2790 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)(8,12)(9,10)(13,14,17,18)(15,19,20,16)(21,22,25,26)(23,27,28,24);(6,10)(7,12)(13,15,17,20)(14,16,18,19)(22,26)(23,28);(6,10)(8,11)(14,18)(15,20)(21,23)(22,27)(24,26)(25,28);(1,2)(3,4)(5,7)(6,11)(8,10)(9,12)(13,16)(14,15)(17,19)(18,20);(5,8,9,11)(6,7,10,12)(13,14)(15,16)(17,18)(19,20)(21,24,25,27)(22,28,26,23)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,e^c=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 908907);