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×#127 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1795 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)(14,18)(15,21)(17,23)(19,27)(22,24)(25,28);(5,7,10,6)(8,11,9,12)(13,14,16,18)(15,19,22,25)(17,20,23,26)(21,28,24,27);(13,15,16,22)(14,19,18,25)(17,24,23,21)(20,27,26,28);(5,8,10,9)(6,11,7,12)(14,18)(19,25)(20,26)(27,28);(1,3,4,2)(5,9,10,8)(6,11,7,12)(14,18)(19,25)(20,26)(27,28)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=i,d^2=fi,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, 6307925);