Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #451783 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16014 |
| 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 | |
1002 conjugacy classes of subgroups; only the 168 normal subgroups are shown.
This group has 1002 conjugacy classes of subgroups — too many to draw. Only its 168 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:
(1,2,6,11,7,12,5,9)(3,13,15,8,17,24,16,25)(4,18,21,10,22,26,20,28)(14,23,19,29,31,32,30,27);(1,3)(2,8)(4,19)(5,16)(6,15)(7,17)(9,13)(10,27)(11,24)(12,25)(14,20)(18,32)(21,31)(22,30)(23,26)(28,29);(2,9)(5,6)(8,13)(10,26)(11,12)(15,16)(18,28)(19,30)(20,21)(23,29)(24,25)(27,32);(1,4)(2,10,12,28)(3,14)(5,21)(6,20)(7,22)(8,23,25,32)(9,26,11,18)(13,29,24,27)(15,30)(16,19)(17,31)Polycyclic presentation:
2^9|a^2=gh,d^2=ef,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,e^b=ei,f^b=fi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72237051, #72237054, #72237078, … |
| maximal normal subgroup of | #72237051, #72237054, #72237078, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 358979);