Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1258513 |
| automorphism | ? |
| 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 | #15124 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2718 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)(3,8)(4,6)(5,7)(9,10)(11,14)(12,15)(13,16)(17,18,22,23)(19,21,24,20)(25,26,28,30)(27,31,32,29);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,16)(14,15)(17,19)(18,20)(21,23)(22,24);(9,10)(11,14)(12,15)(13,16)(17,20,22,21)(18,19,23,24)(25,26)(27,29)(28,30)(31,32);(1,2)(3,6)(4,8)(5,7)(9,10)(11,13)(12,15)(14,16)(17,21,22,20)(18,19,23,24)(25,27,28,32)(26,29,30,31);(1,4,5,3)(2,6,7,8)(9,11)(10,14)(12,16)(13,15)(17,19)(18,21)(20,23)(22,24)(25,26)(27,29)(28,30)(31,32)Polycyclic presentation:
2^9|a^2=i,b^2=fh,c^2=gi,d^2=i,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef86 conjugacy classes — view the full character table.
| maximal subgroup of | #72303502, #72303520, #72303524, … |
| maximal normal subgroup of | #72303502, #72303520, #72303524, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1233633);