Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #479660 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17223 |
| 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 | |
836 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 836 conjugacy classes of subgroups — too many to draw. Only its 152 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)(3,7)(4,8)(5,16,19,9)(6,10)(11,27)(12,25,17,21)(13,22)(14,26,18,23)(15,24)(20,29)(28,31,30,32);(1,3)(2,7)(4,11)(5,17,19,12)(6,13)(8,20)(9,25,16,21)(10,22)(14,28,18,30)(15,29)(23,31,26,32)(24,27);(1,4)(2,8)(3,11)(5,18,19,14)(6,15)(7,20)(9,26,16,23)(10,24)(12,30,17,28)(13,29)(21,32,25,31)(22,27);(1,5)(2,9)(3,12)(4,14)(6,19)(7,21)(8,23)(10,16)(11,28)(13,17)(15,18)(20,31)(22,25)(24,26)(27,32)(29,30)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=g,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=def,f^b=fi,d^c=de,e^c=ei,g^d=gi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72237952, #72237963, #72240456, … |
| maximal normal subgroup of | #72237952, #72237963, #72240456, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 386856);