Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509081 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
960 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)(9,10,12,16,11,19,20,24,14,18,23,15,13,17,22,21);(5,6)(7,8)(9,10)(11,17)(12,21)(13,19)(14,18)(15,20)(16,22)(23,24);(2,4)(6,8);(2,4)(9,11)(10,15)(13,14)(16,18)(17,21)(19,24)(20,22)Polycyclic presentation:
2^9|a^2=ef,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,e^b=eh,h^b=hi,e^d=eh,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72343483, #72343697, #72343816, … |
| maximal normal subgroup of | #72343483, #72343697, #72343816, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 417036);