Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6631114 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C24×C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #23428 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1998 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,22)(20,23)(21,24)(25,28)(26,30)(27,29)(31,32);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(17,19,21,26)(18,22,24,30)(20,27,28,32)(23,29,25,31);(3,7)(6,8)(9,10)(11,15)(12,14)(13,16)(18,23)(20,28)(22,29)(24,25)(27,32)(30,31);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(19,26)(22,30)(27,32)(29,31);(18,24)(22,30)(23,25)(29,31)Polycyclic presentation:
2^9|b^2=hi,d^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,g^a=gi,c^b=ch,d^b=di,d^c=dfh,g^c=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72281724, #72282335, #72282511 |
| maximal normal subgroup of | #72281724, #72282335, #72282511 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6538684);