Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437537 |
| 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 |
| prime radical | C2×#3484 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5436 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,3,4)(9,10,13,18,14,17,12,16)(11,19,22,15,20,23,21,24);(2,4)(5,6)(7,8)(9,11)(10,15)(12,22)(13,21)(14,20)(16,23)(17,24)(18,19);(6,8)(10,16)(12,13)(15,23)(17,18)(19,24)(21,22);(6,8);(10,17)(11,20)(16,18)(21,22)Polycyclic presentation:
2^9|a^2=fi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72312803, #72312809, #72312854, … |
| maximal normal subgroup of | #72312803, #72312809, #72312854, … |
| maximal quotient of | #72315724 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6345063);