Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #424595 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2477 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15371 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
856 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)(3,13,16,8)(4,17)(5,12)(6,11)(7,10)(9,18)(14,22,20,25)(15,29,31,32)(19,26)(21,24)(23,28,27,30);(3,14,16,20)(8,22,13,25)(15,30,31,28)(23,29,27,32);(1,3)(2,8)(4,15)(5,20)(6,14)(7,16)(9,23)(10,13)(11,25)(12,22)(17,32)(18,28)(19,31)(21,30)(24,27)(26,29);(1,4)(2,9,10,24)(3,15)(5,21)(6,18)(7,19)(8,23,13,27)(11,26,12,17)(14,28)(16,31)(20,30)(22,32,25,29)Polycyclic presentation:
2^9|a^2=e,b^2=e,d^2=f,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^c=ei,g^c=gi,g^d=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72319788, #72338619, #72338620, … |
| maximal normal subgroup of | #72319788, #72338619, #72338620, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 334863);