Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395190 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2431 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14893 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
991 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,10)(3,12,15,7)(4,9)(6,11,13,20)(8,19)(14,22)(16,28,29,23)(17,25)(18,26)(21,30)(24,32)(27,31);(1,3,13,29,5,15,6,16)(2,7,20,28,10,12,11,23)(4,14,17,30,18,31,19,32)(8,24,9,22,25,21,26,27);(1,4)(2,8)(3,14)(5,18)(6,19)(7,21)(9,20)(10,25)(11,26)(12,24)(13,17)(15,31)(16,32)(22,23)(27,28)(29,30);(1,4)(2,9,10,26)(3,14)(5,18)(6,19)(7,22,12,27)(8,20,25,11)(13,17)(15,31)(16,32)(21,23,24,28)(29,30)Polycyclic presentation:
2^9|a^2=e,b^2=ef,d^2=g,f^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,h^b=hi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72280350 |
| maximal normal subgroup of | #72280350 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 311728);