Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33605 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×D8 | C23 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
598 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(14,18)(15,19)(16,22)(20,24)(21,28)(23,27)(25,26);(1,3,4,2)(6,9)(10,12);(13,14,19,26,16,21,23,24,17,20,27,28,22,25,15,18);(5,7,8,11)(6,10,9,12)(14,20)(18,24)(21,25)(26,28)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=fh,d^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^c=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72343467, #72343471, #72343483, … |
| maximal normal subgroup of | #72343467, #72343471, #72343483, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402503);