Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667900 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #37648 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2089 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,14)(13,16)(18,22)(20,27)(23,29)(24,28)(26,31)(30,32);(2,5)(3,7)(9,11)(10,13)(12,16)(14,15);(11,16)(13,15)(17,18,21,24)(19,23,25,30)(20,28,27,22)(26,32,31,29);(2,5)(6,8)(10,14)(13,15)(17,19)(18,23)(20,26)(21,25)(22,29)(24,30)(27,31)(28,32);(1,3,4,7)(2,6,5,8)(19,25)(23,30)(26,31)(29,32)Polycyclic presentation:
2^9|c^2=i,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
| maximal subgroup of | #72280978, #72281331, #72281454 |
| maximal normal subgroup of | #72280978, #72281331, #72281454 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6699648);