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×#472 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1540 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,3,4)(5,6)(7,8)(9,10,15,16,14,18,13,12)(11,19,20,21,23,17,22,24);(10,16)(11,20)(12,18)(13,15)(17,19)(22,23);(2,4)(9,11)(10,17)(12,21)(13,22)(14,23)(15,20)(16,24)(18,19);(6,8)(9,12,15,10,14,16,13,18)(11,21,20,17,23,24,22,19)Polycyclic presentation:
2^9|a^2=ef,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 | #72322984 |
| maximal normal subgroup of | #72322984 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 327242);