Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #410821 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15091 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1243 conjugacy classes of subgroups; only the 232 normal subgroups are shown.
This group has 1243 conjugacy classes of subgroups — too many to draw. Only its 232 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,7)(3,12)(4,16)(5,8)(9,19)(10,20)(23,30)(28,31);(1,2,8,11,6,7,5,18)(3,13,21,19,12,25,15,9)(4,10,22,24,16,20,17,29)(14,27,31,30,26,32,28,23);(1,3,5,15,6,12,8,21)(2,9,11,13,7,19,18,25)(4,14,17,28,16,26,22,31)(10,23,24,27,20,30,29,32);(1,4)(2,10)(3,14)(5,17)(6,16)(7,20)(8,22)(9,23)(11,24)(12,26)(13,27)(15,28)(18,29)(19,30)(21,31)(25,32)Polycyclic presentation:
2^9|b^2=gh,c^2=g,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,f^c=fi,e^d=ei65 conjugacy classes — view the full character table.
| maximal subgroup of | #72338524, #72338528, #72338529 |
| maximal normal subgroup of | #72338524, #72338528, #72338529 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 318017);