Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3322 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,3)(9,10,14,18)(11,16,20,23)(12,17,13,22)(15,21,24,19);(1,2,4,3)(5,6)(7,8);(1,3,4,2)(6,8)(9,11)(10,15)(12,19)(13,21)(14,20)(16,22)(17,23)(18,24);(6,8)(9,11)(10,16)(12,21)(13,19)(14,20)(15,22)(17,24)(18,23);(9,12,14,13)(10,17,18,22)(11,19,20,21)(15,23,24,16)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,g^c=gi,e^d=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72316419, #72316462, #72316463 |
| maximal normal subgroup of | #72316419, #72316462, #72316463 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6346005);