Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15257 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2819 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,6)(4,5)(7,8)(9,10,12,14)(11,13,15,16)(18,22)(19,23)(25,26)(27,31)(28,32)(29,30);(3,7)(6,8)(9,11,12,15)(10,13,14,16)(17,18,21,22)(19,20,23,24)(25,27)(26,28)(29,32)(30,31);(2,5)(6,8)(10,14)(13,16)(18,22)(20,24)(26,30)(28,31);(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,19)(18,20)(21,23)(22,24)(25,26)(27,28)(29,30)(31,32);(1,3)(2,6)(4,7)(5,8)(9,11)(10,13)(12,15)(14,16)(17,20)(18,19)(21,24)(22,23)(25,28)(26,27)(29,31)(30,32)Polycyclic presentation:
2^9|a^2=hi,b^2=i,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
| maximal subgroup of | #72300944, #72328672, #72328685, … |
| maximal normal subgroup of | #72300944, #72328672, #72328685, … |
| maximal quotient of | #72331425 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5393633);