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)(7,17)(8,14,9,12)(10,24,25,15)(11,16)(13,20,30,19)(18,28,26,36)(21,31)(22,27)(23,32)(29,34)(33,35);(6,12)(7,18)(10,25)(13,28)(14,16)(15,32)(17,29)(19,34)(20,35)(21,26)(22,23)(24,27)(30,36)(31,33);(2,4)(5,7,8,19)(6,13,12,17)(9,20,11,21)(10,26,22,35)(14,31,16,30)(15,33,27,36)(18,23,34,25)(24,29,32,28)Polycyclic presentation:
2^9|a^2=g,c^2=d,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,h^a=hi,c^b=cf,e^b=eh,f^b=fi,g^b=gh,e^c=ehi,f^c=fi,g^c=ghi,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 | #72312053, #72312071, #72312079, … |
| maximal normal subgroup of | #72312053, #72312071, #72312079, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55667);