Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437510 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4897 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,3,4)(9,10,12,18,14,17,13,15)(11,19,20,16,22,23,21,24);(2,4)(5,6)(7,8)(10,15)(12,13)(16,23)(17,18)(19,24)(20,21);(6,8)(9,11,14,22)(10,16,17,24)(12,21,13,20)(15,23,18,19);(6,8);(10,17)(15,18)(16,24)(19,23)Polycyclic presentation:
2^9|a^2=f,c^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,g^b=gi,f^c=fi,g^c=gi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72308238, #72308365, #72312818, … |
| maximal normal subgroup of | #72308238, #72308365, #72312818, … |
| maximal quotient of | #72315818 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6345036);