Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13318 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1999 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
702 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)(10,28)(11,16)(12,20)(13,24)(15,23)(18,32)(19,35)(21,36)(22,34)(25,33)(27,31)(29,30);(7,18,19,22)(8,9)(10,23)(12,30,31,36)(13,33)(14,26)(15,28)(17,34,35,32)(20,21,27,29)(24,25);(5,7)(6,12)(8,22)(9,27)(10,29)(11,19)(13,30)(14,17)(15,34)(16,31)(18,25)(20,24)(21,23)(26,36)(28,32)(33,35)Polycyclic presentation:
2^9|a^2=g,b^2=d,e^2=hi,f^2=h,h^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fh,h^a=hi,c^b=cf,e^b=ehi,f^b=fh,h^b=hi,d^c=di,e^c=eh,f^c=fhi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250626, #72250642, #72250651, … |
| maximal normal subgroup of | #72250626, #72250642, #72250651, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55314);