Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454025 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1355 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(6,10)(7,8)(13,14,19,27,17,20,18,22)(15,23,26,21,16,25,24,28);(1,2,3,4)(5,6,9,10)(7,11,8,12)(14,20)(21,28)(22,27)(23,25);(2,4)(5,7)(6,11)(8,9)(10,12)(13,15)(14,21)(16,17)(18,26)(19,24)(20,28)(22,25)(23,27);(5,8,9,7)(6,12,10,11)(13,16,17,15)(14,21,20,28)(18,26,19,24)(22,23,27,25)Polycyclic presentation:
2^9|a^2=ef,b^2=egi,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^c=de,f^c=fi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 372014);