Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61643 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#496 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1866 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(5,6)(7,10)(8,12)(9,11)(13,14)(15,22)(16,20)(17,21)(18,19)(23,27)(24,26)(25,28);(5,7,10,6)(8,12,9,11)(13,15,17,18)(14,19,21,22)(16,24,25,23)(20,27,28,26);(5,8,10,9)(6,11,7,12)(14,19)(15,18)(20,26)(21,22)(23,24)(27,28);(5,8,10,9)(6,12,7,11)(13,16,17,25)(14,20,21,28)(15,23,18,24)(19,26,22,27);(1,3,4,2)(5,9,10,8)(6,12,7,11)(14,21)(16,25)(19,22)(23,24)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=fi,d^2=f,e^2=fhi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ci,d^b=df,g^c=gi,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6308201);