Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #116022 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1439 |
| commutator | C25 | C22×C4 |
| Frattini, cosocle | C26 | C23 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C26 ◁ G | |
| LECS | G ▷ C26 ▷ C22 ▷ C1 | |
1509 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,14)(12,19,20,15)(13,16)(17,21,24,23)(18,22);(1,2)(3,7,8,5)(4,6)(9,11)(10,14)(12,17)(13,18)(15,23)(16,22)(19,21)(20,24);(1,3)(2,5)(4,8)(6,7)(9,12)(10,15)(11,17)(13,20)(14,21)(16,19)(18,24)(22,23)Polycyclic presentation:
2^9|a^2=g,b^2=d,b^a=bd,c^a=ce,e^a=ei,f^a=fhi,c^b=cf,e^b=ei,f^b=fh,d^c=dh,g^c=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72233056, #72233066, #72233067, … |
| maximal normal subgroup of | #72233056, #72233066, #72233067, … |
| maximal quotient of | #72264957, #72265906, #72338812 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 23218);