Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417980 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2670 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15404 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
987 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)(3,8)(4,16,18,26)(5,21)(6,10)(7,11)(9,19,25,20)(12,27,17,31)(13,30)(14,23)(15,24)(22,28,32,29);(1,3,6,14,7,15,21,30)(2,8,10,23,11,24,5,13)(4,17,19,29,18,12,20,28)(9,22,16,27,25,32,26,31);(1,4)(2,9)(3,12)(5,16)(6,20)(7,18)(8,22)(10,26)(11,25)(13,27)(14,29)(15,17)(19,21)(23,31)(24,32)(28,30);(1,5,6,2,7,10,21,11)(3,13,14,8,15,23,30,24)(4,16,20,9,18,26,19,25)(12,27,29,22,17,31,28,32)Polycyclic presentation:
2^9|a^2=e,b^2=f,d^2=ef,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,g^b=gi,e^c=ei,f^c=fi,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72338480, #72338482, #72338492 |
| maximal normal subgroup of | #72338480, #72338482, #72338492 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 336771);