Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417575 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3153 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1767 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(6,9)(10,12)(13,14)(15,19)(16,25)(17,22)(18,21)(20,24)(23,28)(26,27);(1,2)(3,4)(5,6,8,9)(7,10,11,12)(13,15,17,18)(14,19,22,21)(16,24,26,23)(20,27,28,25);(2,4)(5,7,8,11)(6,10,9,12)(13,16)(14,20)(15,23)(17,26)(18,24)(19,25)(21,27)(22,28);(6,9)(10,12)(13,16)(14,20)(15,24)(17,26)(18,23)(19,27)(21,25)(22,28)Polycyclic presentation:
2^9|b^2=ei,c^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,f^c=fi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 325357);