Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13472 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2067 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
730 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6,10,14,11,17,8,24)(7,18,22,33,23,32,20,12)(9,26,28,16,29,13,25,15)(19,34,21,30,27,35,36,31);(1,2,3,4)(5,7,8,20,11,23,10,22)(6,12,14,18,17,33,24,32)(9,27,25,21,29,19,28,36)(13,31,15,34,26,30,16,35);(6,13)(7,19)(8,10)(9,28)(14,16)(15,24)(17,26)(18,32)(20,21)(22,36)(23,27)(25,29)(30,35)(31,34)Polycyclic presentation:
2^9|a^2=g,b^2=d,d^2=i,e^2=h,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,f^a=fhi,h^a=hi,c^b=cf,e^b=eh,f^b=fh,h^b=hi,d^c=di,e^c=ehi,f^c=fhi,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250133, #72250141, #72250147, … |
| maximal normal subgroup of | #72250133, #72250141, #72250147, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55345);