Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24404 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
2792 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)(15,21,17,19)(16,20)(18,22)(23,26,24,27)(25,28);(5,6,8,10)(7,11,12,9)(13,15,18,17)(14,19,22,21)(16,24,25,23)(20,27,28,26);(2,4)(5,7,8,12)(6,9,10,11)(13,16,18,25)(14,20,22,28)(15,23,17,24)(19,26,21,27);(13,17,18,15)(14,21,22,19)(16,24,25,23)(20,27,28,26)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,d^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,f^b=fi,h^b=hi,e^c=ei,f^d=fi,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72268067, #72268069, #72268102, … |
| maximal normal subgroup of | #72268067, #72268069, #72268102, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 269937);