Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #246987 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #51902 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1321 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 1321 conjugacy classes of subgroups — too many to draw. Only its 150 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)(3,7,14,22)(4,8,17,25)(5,11,18,29)(9,26,27,13)(10,19,24,16)(15,28,31,20)(21,30,23,32);(2,7)(3,13)(4,16)(8,21)(9,11)(10,28)(12,22)(14,26)(15,31)(17,19)(20,24)(23,25)(27,29)(30,32);(2,8)(3,14)(4,16)(7,20)(9,21)(10,29)(11,24)(12,25)(13,26)(15,30)(17,19)(22,28)(23,27)(31,32);(2,7)(3,13)(4,17)(8,20)(9,11)(10,23)(12,22)(14,26)(15,32)(16,19)(21,24)(25,28)(27,29)(30,31)Polycyclic presentation:
2^9|a^2=i,e^2=h,f^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fhi,g^a=gh,c^b=cg,e^b=eh,f^b=fhi,e^c=ei,f^c=fhi,e^d=eh,f^d=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72213691, #72219510, #72219511, … |
| maximal normal subgroup of | #72213691, #72219510, #72219511, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 154189);