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 | C22×#575 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4846 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)(9,10)(11,17)(12,23)(13,18)(14,20)(15,19)(16,21)(22,24);(5,6)(7,8)(9,11)(10,16)(12,13)(14,21)(15,22)(17,18)(19,23)(20,24);(2,4)(6,8)(9,12,13,11,15,21,14,22)(10,17,20,23,19,24,18,16);(9,13,15,14)(10,18,19,20)(11,12,22,21)(16,17,23,24);(10,19)(16,23)(17,24)(18,20)Polycyclic presentation:
2^9|c^2=f,d^2=i,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=ch,f^b=fi,g^b=gi,h^b=hi,d^c=di92 conjugacy classes — view the full character table.
| maximal subgroup of | #72249548, #72249569, #72249572, … |
| maximal normal subgroup of | #72249548, #72249569, #72249572, … |
| maximal quotient of | #72264900, #72310691, C2×#6431967, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6339163);