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) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15278 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2182 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,3)(4,7,8,6)(9,10)(11,15)(12,14)(13,16)(18,22)(19,24)(25,26,29,31)(27,28,32,30);(1,2,5,3)(4,7,8,6)(9,11)(10,15)(12,13)(14,16)(17,18,21,22)(19,23,24,20)(25,27,29,32)(26,30,31,28);(2,3)(6,7)(10,12)(11,16)(17,19,21,24)(18,23,22,20)(25,26,29,31)(27,30,32,28);(1,3,5,2)(4,7,8,6)(9,12,14,10)(11,13,16,15)(17,20,21,23)(18,19,22,24)(25,28)(26,27)(29,30)(31,32);(1,4)(2,6)(3,7)(5,8)(9,13)(10,11)(12,16)(14,15)(17,19)(18,23)(20,22)(21,24)(25,26)(27,28)(29,31)(30,32)Polycyclic presentation:
2^9|a^2=hi,b^2=gi,c^2=f,d^2=g,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.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5393701);