Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1418353 |
| 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 | C22×#121 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2570 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)(9,10,14,15)(11,12,16,13)(17,18,21,20)(19,23,24,22)(25,26,29,32)(27,30,31,28);(2,6)(7,8)(9,11,14,16)(10,13,15,12)(25,27,29,31)(26,30,32,28);(1,3,5,4)(2,7,6,8)(9,12,14,13)(10,11,15,16)(17,19)(18,22)(20,23)(21,24)(25,26)(27,30)(28,31)(29,32);(1,4,5,3)(2,7,6,8)(9,13,14,12)(10,11,15,16)(17,20,21,18)(19,23,24,22)(25,28,29,30)(26,31,32,27);(1,2)(3,7)(4,8)(5,6)(9,11)(10,13)(12,15)(14,16)(17,19)(18,22)(20,23)(21,24)(25,26)(27,30)(28,31)(29,32)Polycyclic presentation:
2^9|a^2=g,b^2=fi,c^2=f,d^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,e^d=ef92 conjugacy classes — view the full character table.
| maximal subgroup of | #72303399, #72303442, #72303515, … |
| maximal normal subgroup of | #72303399, #72303442, #72303515, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1345324);