Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #420461 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #35528 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1699 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,6)(7,8)(9,10,12,21,13,17,23,18)(11,16,19,15,20,14,22,24);(1,2,5,6)(3,7,4,8)(9,11)(10,14)(12,22)(13,20)(15,21)(16,17)(18,24)(19,23);(1,3,5,4)(2,7,6,8)(9,11)(10,15,17,24)(12,19)(13,20)(14,18,16,21)(22,23);(1,4,5,3)(2,7,6,8)(9,11)(10,16,17,14)(12,22)(13,20)(15,21,24,18)(19,23)Polycyclic presentation:
2^9|a^2=fg,b^2=eg,c^2=e,d^2=e,f^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,f^b=fi,h^b=hi,h^c=hi,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72252500, #72252503, #72252506, … |
| maximal normal subgroup of | #72252500, #72252503, #72252506, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 348303);