Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907993 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1947 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(3,7)(17,18)(19,20)(21,24)(22,28)(23,25)(26,27)(29,31)(30,32);(1,2,4,5)(3,6,7,8)(9,10)(11,15)(12,14)(13,16)(17,19,23,22)(18,20,25,28)(21,26,30,31)(24,27,32,29);(3,7)(6,8)(9,11)(10,13)(12,16)(14,15)(17,20,19,18,23,28,22,25)(21,29,26,32,30,27,31,24);(1,3,4,7)(2,6,5,8)(11,16)(13,15)(17,21)(18,24)(19,26)(20,27)(22,31)(23,30)(25,32)(28,29);(2,5)(6,8)(10,14)(13,15)(21,30)(24,32)(26,31)(27,29)Polycyclic presentation:
2^9|b^2=fi,c^2=gi,d^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,d^c=dfh,e^d=ei62 conjugacy classes — view the full character table.
| maximal subgroup of | #72308665, #72308877 |
| maximal normal subgroup of | #72308665, #72308877 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6817425);