Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6431707 |
| 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×#3449 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4882 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)(11,12)(13,17)(14,18)(15,19)(16,22)(20,24)(21,23);(5,6)(7,8)(9,11)(10,12)(13,20)(14,16)(15,21)(17,24)(18,22)(19,23);(2,4)(5,7,8,6)(9,12)(10,16)(11,18)(13,22)(14,24)(15,23)(17,21)(19,20);(9,13,15,14)(10,17,19,18)(11,16,21,20)(12,22,23,24);(9,13,15,14)(10,18,19,17)(11,20,21,16)(12,22,23,24)Polycyclic presentation:
2^9|c^2=f,d^2=i,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,h^a=hi,c^b=ch,d^b=di,g^b=gi,h^b=hi,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72305326, #72305336, #72305337, … |
| maximal normal subgroup of | #72305326, #72305336, #72305337, … |
| maximal quotient of | #72306016, #72316005, #72316014 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6339105);