Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5020720 |
| automorphism | Aut(G) order 9437184 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C27 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
9346 conjugacy classes of subgroups; only the 538 normal subgroups are shown.
This group has 9346 conjugacy classes of subgroups — too many to draw. Only its 538 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 — 5 generators on 32 points:
(1,2,5,6)(3,4,7,8)(9,10,13,14)(11,15,16,12)(17,18,21,22)(19,23,24,20)(25,26,29,30)(27,28,31,32);(1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,16)(14,15)(17,19)(18,20)(21,24)(22,23)(25,27)(26,28)(29,31)(30,32);(2,6)(3,7)(10,14)(12,15)(17,19,21,24)(18,20,22,23)(25,28,29,32)(26,27,30,31);(1,3)(2,4)(5,7)(6,8)(9,12)(10,11)(13,15)(14,16)(17,19)(18,20)(21,24)(22,23)(25,28)(26,27)(29,32)(30,31);(1,4)(2,3)(5,8)(6,7)(9,10)(11,12)(13,14)(15,16)(17,20)(18,19)(21,23)(22,24)(25,26)(27,28)(29,30)(31,32)Polycyclic presentation:
2^9|a^2=g,c^2=i,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^c=dg,e^c=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72219200, #72219202, #72241210, … |
| maximal normal subgroup of | #72219200, #72219202, #72241210, … |
| maximal quotient of | #72219193, #72219194 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4931699);