Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454025 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1071 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6,10,8)(7,12,9,11)(13,14,16,19)(15,23,24,28)(17,22,18,21)(20,26,27,25);(1,2,3,4)(5,7,10,9)(6,11,8,12)(14,19)(20,27)(21,22)(23,28);(2,4)(5,8,10,6)(7,11,9,12)(13,15,16,24)(14,20,19,27)(17,26,18,25)(21,28,22,23);(5,9,10,7)(6,12,8,11)(13,15)(14,20)(16,24)(17,25)(18,26)(19,27)(21,23)(22,28)Polycyclic presentation:
2^9|a^2=e,b^2=g,c^2=e,d^2=ei,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^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, 371709);