Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #439378 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15636 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
982 conjugacy classes of subgroups; only the 184 normal subgroups are shown.
This group has 982 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)(3,11,14,7)(4,8)(5,17)(6,10)(9,18)(12,19,26,21)(13,24,27,20)(15,28)(16,23)(22,29)(25,30,32,31);(1,3)(2,7)(4,12,16,26)(5,13)(6,14)(8,21,23,19)(9,24)(10,11)(15,25,29,32)(17,20)(18,27)(22,30,28,31);(1,4)(2,8)(3,12,14,26)(5,15)(6,16)(7,19,11,21)(9,22)(10,23)(13,25,27,32)(17,28)(18,29)(20,30,24,31);(1,5)(2,9,10,17)(3,13)(4,15)(6,18)(7,20,11,24)(8,22,23,28)(12,25)(14,27)(16,29)(19,30,21,31)(26,32)Polycyclic presentation:
2^9|a^2=f,b^2=efg,c^2=f,d^2=e,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72237785, #72237786, #72237793, … |
| maximal normal subgroup of | #72237785, #72237786, #72237793, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 346574);