Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61629 |
| 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×#550 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1925 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)(8,11)(9,12)(13,14)(15,20)(16,26)(17,23)(18,22)(19,24)(21,27)(25,28);(5,7,10,6)(8,11,9,12)(13,15,19,16)(14,20,24,26)(17,27,18,25)(21,22,28,23);(13,16,19,15)(14,21,24,28)(17,27,18,25)(20,22,26,23);(5,8,10,9)(6,11,7,12);(1,3,4,2)(5,9,10,8)(6,11,7,12)(13,17,19,18)(14,22,24,23)(15,25,16,27)(20,21,26,28)Polycyclic presentation:
2^9|a^2=h,b^2=fi,c^2=i,d^2=f,e^2=fhi,g^2=i,b^a=bf,c^a=cg,e^a=efi,g^a=gi,d^b=df,e^b=ei,g^c=gi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6307921);