Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806903 |
| 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 ◁ C4×#120 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2255 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,14)(15,20)(16,21)(17,22)(18,26)(19,23)(24,27)(25,28);(5,7)(6,9)(8,12)(10,11)(13,15)(14,20)(16,25)(17,18)(19,24)(21,28)(22,26)(23,27);(2,4)(7,12)(9,11)(13,16,25,15,19,18,17,24)(14,21,28,20,23,26,22,27);(6,10)(9,11)(13,17,19,25)(14,22,23,28)(15,18,24,16)(20,26,27,21);(13,18,25,24,19,16,17,15)(14,21,28,20,23,26,22,27)Polycyclic presentation:
2^9|c^2=fhi,d^2=i,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6717346);