Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23897 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2403 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14127 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2172 conjugacy classes of subgroups; only the 344 normal subgroups are shown.
This group has 2172 conjugacy classes of subgroups — too many to draw. Only its 344 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:
(1,2,6,10,7,11,5,18)(3,12,15,8,14,23,19,21)(4,9,16,26,17,25,20,24)(13,28,31,32,29,27,30,22);(8,21)(12,23)(22,32)(27,28);(1,3,6,15,7,14,5,19)(2,8,10,23,11,21,18,12)(4,13,16,31,17,29,20,30)(9,22,26,28,25,32,24,27);(1,4)(2,9)(3,13)(5,20)(6,16)(7,17)(8,22)(10,26)(11,25)(12,27)(14,29)(15,31)(18,24)(19,30)(21,32)(23,28)Polycyclic presentation:
2^9|a^2=h,c^2=h,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,e^c=ei,f^c=fi,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72286057, #72286087, #72286225, … |
| maximal normal subgroup of | #72286057, #72286087, #72286225, … |
| maximal quotient of | C2×#373643 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 280839);