Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13595 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#354 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | #127 | C24 |
| socle | C2 | C22×#354 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×C4 ◁ C2×#532 ◁ G | |
| LECS | G ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
845 conjugacy classes of subgroups; only the 192 normal subgroups are shown.
This group has 845 conjugacy classes of subgroups — too many to draw. Only its 192 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)(3,12,15,20)(4,8)(5,10,19,9)(6,11)(7,13,22,18)(14,26,28,31)(16,24,30,23)(17,25)(21,27,32,29);(1,3,6,13)(2,7)(4,14,17,27)(5,18,19,15)(8,21)(9,12)(10,22)(11,20)(16,29,30,28)(23,26)(24,32)(25,31);(3,13)(7,20)(12,22)(14,27)(15,18)(21,31)(26,32)(28,29);(1,4,6,17)(2,8,11,25)(3,14,13,27)(5,16,19,30)(7,21,20,31)(9,23,10,24)(12,26,22,32)(15,28,18,29)Polycyclic presentation:
2^9|a^2=f,b^2=h,d^2=i,e^2=hi,g^2=i,b^a=be,e^a=eg,g^a=gh,h^a=hi,c^b=ci,e^b=eh,f^b=fgh,g^b=gi,f^e=fhi,g^f=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72279345, #72279362, #72279397, … |
| maximal normal subgroup of | #72279345, #72279362, #72279397, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419821);