Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573523 |
| 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 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4804 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)(5,6)(7,8)(11,18)(15,20)(19,24)(21,23);(2,4)(10,12)(11,18)(13,16)(15,24)(17,22)(19,20);(6,8)(9,10)(11,19)(12,16)(13,22)(14,17)(15,23)(18,24)(20,21);(9,11,14,18)(10,15,17,20)(12,19,22,24)(13,23,16,21);(9,12,14,22)(10,16,17,13)(11,20,18,15)(19,21,24,23)Polycyclic presentation:
2^9|d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72308549, #72308908 |
| maximal normal subgroup of | #72308549, #72308908 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6480725);