Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23876 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2514 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,3,4)(13,14)(15,25)(16,20)(17,23)(18,22)(19,24)(21,26)(27,28);(5,6,8,10)(7,11,12,9)(13,15,19,26)(14,20,24,28)(16,17,27,18)(21,23,25,22);(5,7,8,12)(6,9,10,11)(13,16)(14,21)(15,18)(17,26)(19,27)(20,22)(23,28)(24,25);(13,16)(14,21)(15,17)(18,26)(19,27)(20,23)(22,28)(24,25)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=gi,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^c=fi,e^d=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72268165, #72268184, #72271418, … |
| maximal normal subgroup of | #72268165, #72268184, #72271418, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 269982);