Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1418353 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1883 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,6)(3,4,7,8)(9,10,14,15)(11,13,12,16)(19,24)(20,22)(26,29)(27,31);(2,6)(4,8)(10,15)(13,16)(17,18,21,23)(19,22,24,20)(26,29)(30,32);(9,11,14,12)(10,13,15,16)(17,19)(18,22)(20,23)(21,24)(25,26,28,29)(27,30,31,32);(1,3)(2,4)(5,7)(6,8)(9,12,14,11)(10,13,15,16)(17,19)(18,22)(20,23)(21,24);(1,4)(2,7)(3,6)(5,8)(9,13)(10,12)(11,15)(14,16)(17,20,21,22)(18,19,23,24)(25,27,28,31)(26,30,29,32)Polycyclic presentation:
2^9|a^2=f,b^2=gh,c^2=gi,d^2=ghi,e^2=g,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.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1406751);