Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417647 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#459 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1596 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(5,6)(7,8)(9,10,13,12,14,17,22,15)(11,16,20,19,21,24,18,23);(10,15)(11,18)(12,17)(13,22)(16,24)(20,21);(1,2,3,4)(9,11)(10,16)(12,19)(13,20)(14,21)(15,23)(17,24)(18,22);(6,8)(9,12,22,10,14,15,13,17)(11,19,18,16,21,23,20,24)Polycyclic presentation:
2^9|a^2=e,c^2=f,d^2=ei,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,e^b=ei,g^b=gi,g^c=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, 331020);