Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #346284 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2840 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
961 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 961 conjugacy classes of subgroups — too many to draw. Only its 96 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:
(2,7)(3,12)(4,14)(8,21)(9,11)(13,29)(15,27)(17,23)(19,22)(26,30);(1,2,4,10)(3,13,16,8)(5,7,14,20)(6,11,17,25)(9,23,24,18)(12,26,28,19)(15,29,31,22)(21,27,30,32);(1,3,14,28)(2,8,20,26)(4,16,5,12)(6,15,23,32)(7,19,10,13)(9,21,25,29)(11,22,24,30)(17,31,18,27);(2,9)(3,15)(4,17)(7,11)(10,20)(12,27)(13,29)(14,23)(24,25)(26,30)Polycyclic presentation:
2^9|b^2=g,c^2=gh,b^a=be,c^a=cf,g^a=gh,c^b=cg,d^b=defh,e^b=eh,f^b=fhi,d^c=de,e^c=ei,f^c=fh,g^d=gi32 conjugacy classes — view the full character table.
| maximal subgroup of | #72235515, #72238992, #72242693, … |
| maximal normal subgroup of | #72235515, #72238992, #72242693, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 253483);