Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #348128 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2847 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #29684 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1175 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(5,8)(10,14)(12,20)(15,21)(16,22)(17,24)(19,23);(9,10,12,14,13,16,20,22)(11,17,19,24,18,15,23,21);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15)(12,19)(13,18)(14,21)(16,17)(20,23)(22,24);(1,3)(2,5)(4,7)(6,8)(11,18)(15,17)(19,23)(21,24)Polycyclic presentation:
2^9|b^2=e,e^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,c^b=ch,d^c=dh,g^c=gi,f^d=fi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72339481, #72339492, #72339501, … |
| maximal normal subgroup of | #72339481, #72339492, #72339501, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 255324);