Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418438 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1229 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,4,3)(6,10)(11,12)(14,18)(15,22)(16,20)(19,25)(23,24)(26,28);(2,3)(5,6)(7,11)(8,12)(9,10)(14,19)(18,25)(21,27)(26,28);(1,2,4,3)(5,7,9,8)(6,11,10,12)(13,14,20,25,17,19,16,18)(15,21,22,28,24,27,23,26);(1,3,4,2)(5,8,9,7)(6,11,10,12)(13,15)(14,21)(16,23)(17,24)(18,26)(19,27)(20,22)(25,28)Polycyclic presentation:
2^9|a^2=gi,c^2=efi,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 341583);