Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #58990 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #52117 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2183 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)(6,9)(10,12)(13,14,19,23,18,21,24,27)(15,17,16,26,25,28,22,20);(5,6)(7,10)(8,9)(11,12)(13,15)(14,17)(16,19)(18,25)(20,27)(21,28)(22,24)(23,26);(2,4)(5,7,8,11)(6,10,9,12)(13,16)(14,20)(15,24)(17,23)(18,22)(19,25)(21,26)(27,28);(6,9)(10,12)(14,21)(15,25)(16,22)(23,27);(13,17,19,26,18,28,24,20)(14,22,23,15,21,16,27,25)Polycyclic presentation:
2^9|a^2=h,c^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ci,d^b=dfi,e^b=ei,d^c=di,e^c=ei128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6291384);