Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #213852 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2308 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
1430 conjugacy classes of subgroups; only the 190 normal subgroups are shown.
This group has 1430 conjugacy classes of subgroups — too many to draw. Only its 190 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:
(1,2,4,11)(3,12,15,26)(5,10,18,9)(6,8,17,25)(7,13,21,14)(16,20,30,31)(19,24,29,23)(22,27,32,28);(1,3,6,16)(2,7,8,22)(4,13,17,27)(5,14,19,28)(9,21,23,32)(10,12,24,20)(11,26,25,31)(15,29,30,18);(2,8)(4,17)(7,20)(10,24)(12,22)(13,27)(15,30)(18,29)(21,26)(31,32);(1,4,5,18)(2,9,10,11)(3,13,14,15)(6,17,19,29)(7,21,12,26)(8,23,24,25)(16,27,28,30)(20,31,22,32)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=ce,f^b=fh,g^b=gh,d^c=di,g^c=gi,g^d=gh56 conjugacy classes — view the full character table.
| maximal subgroup of | #72240632, #72240633, #72240637, … |
| maximal normal subgroup of | #72240632, #72240633, #72240637, … |
| maximal quotient of | #72251577, #72251586 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 121054);