Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13360 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2021 |
| commutator | C2×#41 | C22×C4 |
| Frattini, prime radical, cosocle | C22×#41 | C23 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
802 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,3,4)(6,11)(7,15)(8,18)(9,21)(12,26)(13,29)(14,25)(16,33)(17,22)(19,32)(23,27)(24,31)(28,36)(30,35);(5,6,8,11,10,14,18,25)(7,12,16,23,17,28,19,35)(9,13,20,24,21,31,34,29)(15,27,32,26,22,30,33,36);(5,7,10,17)(6,12,14,28)(8,19,18,16)(9,22,21,15)(11,23,25,35)(13,30,31,27)(20,32,34,33)(24,36,29,26)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=i,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fhi,c^b=cf,f^b=fh,h^b=hi,d^c=dh,e^c=ei,f^d=fi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55473);