Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417647 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#447 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2514 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,3,4)(9,10,12,18,14,16,13,17)(11,19,20,15,22,23,21,24);(5,6)(7,8)(9,11,12,20,14,22,13,21)(10,15,17,19,16,24,18,23);(2,4)(6,8)(10,16)(11,20)(12,13)(15,23)(19,24)(21,22);(10,16)(11,21)(12,13)(15,19)(20,22)(23,24)Polycyclic presentation:
2^9|a^2=ef,b^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^c=ei,g^c=gi,e^d=ei,g^d=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72323014 |
| maximal normal subgroup of | #72323014 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326380);