Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13334 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2011 |
| 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 | |
748 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)(9,26)(10,27)(11,16)(12,20)(13,18)(15,24)(19,28)(21,33)(22,30)(23,36)(25,31)(29,35)(32,34);(5,7,18,34,10,23,25,19,11,21,9,22,24,35,8,20)(6,12,14,29,15,30,26,33,16,28,31,36,27,32,13,17);(7,19)(8,18)(9,25)(10,24)(12,17)(13,14)(15,27)(20,21)(22,35)(23,34)(26,31)(28,33)(29,32)(30,36)Polycyclic presentation:
2^9|a^2=g,b^2=dh,d^2=hi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fh,h^a=hi,c^b=cf,e^b=ei,d^c=dh,f^c=fhi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250522, #72250530, #72250542, … |
| maximal normal subgroup of | #72250522, #72250530, #72250542, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55607);