Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13432 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2044 |
| commutator | C2×#41 | C22×C4 |
| Frattini, cosocle | C22×#41 | C23 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
810 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6)(7,16)(8,13,20,11)(9,14,24,32)(10,15)(12,18,29,36)(17,33,22,26)(19,27)(21,30)(23,25)(28,34)(31,35);(6,11)(7,17)(8,20)(12,26)(13,15)(14,25)(16,28)(18,35)(19,22)(21,23)(27,31)(29,33)(30,32)(34,36);(2,4)(5,7,8,18,10,19,20,36)(6,12,13,27,15,29,11,16)(9,22,21,35,24,17,23,34)(14,31,30,26,32,28,25,33)Polycyclic presentation:
2^9|a^2=g,c^2=d,d^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,h^a=hi,c^b=cf,d^b=di,e^b=eh,g^b=ghi,e^c=eh,f^c=fi,g^c=gh,h^c=hi,e^d=ei,g^d=gi,f^e=fi,g^e=gi,g^f=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72312034, #72312079, #72312086, … |
| maximal normal subgroup of | #72312034, #72312079, #72312086, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55670);