Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393958 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3003 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4757 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)(5,6)(7,8)(9,10)(11,15)(12,14)(13,17)(16,20)(18,23)(19,24)(21,22);(10,14)(12,20)(15,23)(16,17)(18,21)(22,24);(2,4)(9,11,13,19)(10,15,17,24)(12,21,20,18)(14,22,16,23);(6,8)Polycyclic presentation:
2^9|c^2=i,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,e^c=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72238106, #72256181, #72280401, … |
| maximal normal subgroup of | #72238106, #72256181, #72280401, … |
| maximal quotient of | #72238109 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301156);