Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432838 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #23391 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2160 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(19,27)(20,21)(23,30)(25,26)(28,29)(31,32);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,15)(14,16)(17,18,22,24)(19,23,28,31)(20,25,21,26)(27,30,29,32);(11,15)(13,16)(17,19)(18,23)(20,27)(21,29)(22,28)(24,31)(25,30)(26,32);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(18,24)(23,31)(25,26)(30,32);(17,20,22,21)(18,25,24,26)(19,27,28,29)(23,30,31,32)Polycyclic presentation:
2^9|b^2=fi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,g^a=gi,c^b=ch,d^b=di,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72282093 |
| maximal normal subgroup of | #72282093 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6340648);