Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417647 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1747 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(6,7)(9,10,12,17,13,16,21,15)(11,14,18,23,20,22,19,24);(5,6)(7,8)(9,11)(10,14)(12,18)(13,20)(15,24)(16,22)(17,23)(19,21);(10,15)(11,18)(12,21)(16,17)(19,20)(23,24);(2,4)(5,7,8,6)(10,16)(14,22)(15,17)(23,24)Polycyclic presentation:
2^9|a^2=f,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,g^b=gi,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72321053, #72321993 |
| maximal normal subgroup of | #72321053, #72321993 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326939);