Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393944 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1318 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,3,4)(9,10,12,13)(11,15,16,14)(17,18)(19,21)(20,24)(22,23);(5,6)(7,8)(10,13)(14,15)(17,19,21,18,22,24,20,23);(2,4)(6,8)(18,19)(20,21)(23,24);(9,11,12,16)(10,14,13,15)(18,23)(19,24)Polycyclic presentation:
2^9|a^2=ef,b^2=eh,d^2=hi,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=di,e^c=ei,f^c=fi,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72264301, #72264315, #72264355, … |
| maximal normal subgroup of | #72264301, #72264315, #72264355, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 323473);