Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #115204 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1438 |
| 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 | |
1835 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)(3,8,6,4)(5,7)(9,10)(11,17,19,14)(12,20,21,15)(13,16)(18,23)(22,24);(1,3,5,6)(2,4,7,8)(9,11)(10,14)(12,18)(13,19)(15,23)(16,17)(20,22)(21,24);(1,4)(2,6)(3,7)(5,8)(9,12)(10,15)(11,18)(13,21)(14,22)(16,20)(17,23)(19,24)Polycyclic presentation:
2^9|a^2=g,b^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fhi,c^b=cf,e^b=eh,g^b=gh,d^c=di,g^c=gh62 conjugacy classes — view the full character table.
| maximal subgroup of | #72233890, #72233894, #72233895, … |
| maximal normal subgroup of | #72233890, #72233894, #72233895, … |
| maximal quotient of | #72236530, #72256711 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 22401);