Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23391 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| prime radical | C22×#532 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5138 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)(6,7)(9,10)(11,19)(12,17)(13,18)(14,16)(15,22)(20,23)(21,24);(5,6)(7,8)(9,10)(11,20)(12,17)(13,18)(14,16)(15,21)(19,23)(22,24);(2,4)(5,7,8,6)(9,11)(10,15)(12,21)(13,22)(14,23)(16,24)(17,19)(18,20);(10,16)(15,24)(17,18)(19,20);(9,12,14,13)(10,17,16,18)(11,21,23,22)(15,19,24,20)Polycyclic presentation:
2^9|c^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,h^a=hi,c^b=ch,d^b=di,g^b=gi,h^b=hi,g^c=gi,h^c=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72307352, #72307372, #72307517, … |
| maximal normal subgroup of | #72307352, #72307372, #72307517, … |
| maximal quotient of | #72316308 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6339101);