Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #346224 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2760 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | #10336 | C2 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
753 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 753 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,9,11,6)(4,12,14,7)(5,8)(10,16)(13,15)(17,18)(19,22,25,30)(20,27,28,23)(21,24)(26,32)(29,31);(1,3)(2,6)(4,13)(5,11)(7,15)(8,9)(10,14)(12,16)(19,25)(29,32);(1,4)(2,7)(3,10,11,13)(5,14)(6,15,9,16)(8,12)(17,19)(18,22,24,30)(20,26)(21,25)(23,29,27,32)(28,31);(6,9)(15,16)(17,20)(18,23)(19,26)(21,28)(22,29)(24,27)(25,31)(30,32)Polycyclic presentation:
2^9|a^2=fgi,c^2=ef,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fhi,g^a=ghi,c^b=cg,d^b=de,f^b=fh,e^c=ei,f^c=fi,g^c=gh,g^d=gi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72241347, #72241358, #72283484, … |
| maximal normal subgroup of | #72241347, #72241358, #72283484, … |
| maximal quotient of | C2×#346239 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 253435);