Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #444622 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15784 |
| 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 | |
1114 conjugacy classes of subgroups; only the 184 normal subgroups are shown.
This group has 1114 conjugacy classes of subgroups — too many to draw. Only its 184 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:
(1,2,6,12,7,11,5,13)(3,14,16,10,17,8,23,25)(4,18,21,28,22,29,20,9)(15,26,19,30,32,27,31,24);(1,3)(2,8)(4,19)(5,16)(6,23)(7,17)(9,26)(10,12)(11,14)(13,25)(15,21)(18,24)(20,32)(22,31)(27,28)(29,30);(1,4,6,21,7,22,5,20)(2,9,13,29,11,28,12,18)(3,15,16,19,17,32,23,31)(8,24,10,27,14,30,25,26);(1,3)(2,10,11,25)(4,15)(5,23)(6,16)(7,17)(8,13,14,12)(9,27,28,26)(18,24,29,30)(19,21)(20,31)(22,32)Polycyclic presentation:
2^9|a^2=g,c^2=eg,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,g^b=gi,h^b=hi,h^c=hi,f^d=fi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72213514, #72216029, #72216843, … |
| maximal normal subgroup of | #72213514, #72216029, #72216843, … |
| maximal quotient of | #72250600, C2×#444622, #72255806, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 351818);