Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23391 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3474 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)(11,20)(12,21)(13,18)(14,19)(15,17)(16,23)(22,24);(5,6)(7,8)(9,11,13,21,15,16,14,22)(10,12,19,20,17,24,18,23);(2,4)(6,8)(9,12)(10,16)(11,17)(13,23)(14,20)(15,24)(18,22)(19,21);(10,17)(11,16)(18,19)(21,22);(9,13,15,14)(10,18,17,19)(11,21,16,22)(12,23,24,20)Polycyclic presentation:
2^9|a^2=fg,b^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,g^a=gi,h^a=hi,c^b=ch,d^b=di,f^c=fi,g^c=gi,h^c=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72307372, #72307559, #72307635, … |
| maximal normal subgroup of | #72307372, #72307559, #72307635, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6339335);