Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437753 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3212 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,8,9)(7,11,12,10)(13,14,19,22)(15,23,24,28)(16,25,27,20)(17,21,18,26);(1,2)(3,4)(6,9)(10,11)(13,15,17,25,19,24,18,20)(14,16,26,28,22,27,21,23);(2,4)(5,7,8,12)(6,10,9,11)(13,16,19,27)(14,20,22,25)(15,21,24,26)(17,28,18,23);(2,4)(15,24)(16,27)(20,25)(23,28);(13,17,19,18)(14,21,22,26)(15,20,24,25)(16,28,27,23)Polycyclic presentation:
2^9|a^2=g,b^2=fg,c^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^b=ei,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6354835);