Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #988785 |
| automorphism | ? |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1758 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)(3,7)(4,8)(5,6)(9,10,14,12)(11,16,15,13)(18,22)(19,24)(25,26,30,29)(27,32,31,28);(1,2,5,6)(3,7,8,4)(9,11)(10,13)(12,16)(14,15)(17,18,21,22)(19,20,24,23)(27,31)(28,32);(2,6)(4,7)(9,10,14,12)(11,16,15,13)(17,19,21,24)(18,23,22,20)(25,27)(26,28)(29,32)(30,31);(1,3)(2,4)(5,8)(6,7)(9,12,14,10)(11,16,15,13)(17,20)(18,19)(21,23)(22,24)(25,28,30,32)(26,31,29,27);(1,4)(2,3)(5,7)(6,8)(9,13,14,16)(10,15,12,11)(17,19)(18,20)(21,24)(22,23)(25,29,30,26)(27,32,31,28)Polycyclic presentation:
2^9|a^2=fg,b^2=fgh,c^2=hi,d^2=fg,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,e^c=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 948024);