Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #469875 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2607 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #16823 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
910 conjugacy classes of subgroups; only the 176 normal subgroups are shown.
This group has 910 conjugacy classes of subgroups — too many to draw. Only its 176 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)(3,11,14,7)(4,8)(5,19)(6,9)(10,12)(13,20,29,25)(15,21,27,26)(16,30,31,22)(17,32)(18,23)(24,28);(1,3,12,27,6,14,5,15)(2,7,10,21,9,11,19,26)(4,16,17,29,18,31,28,13)(8,22,32,20,23,30,24,25);(1,4)(2,8)(3,13,14,29)(5,17)(6,18)(7,20,11,25)(9,23)(10,24)(12,28)(15,16,27,31)(19,32)(21,22,26,30);(2,9)(7,11)(10,19)(21,26)Polycyclic presentation:
2^9|a^2=f,b^2=egh,c^2=f,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,f^b=fi,d^c=de,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72251701, #72251705, #72251712, … |
| maximal normal subgroup of | #72251701, #72251705, #72251712, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 379720);