Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #136733 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1879 |
| 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 | |
736 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6,10,16,11,18,27,17)(7,12,21,31,23,29,22,35)(8,24,28,14,9,13,26,15)(19,32,36,34,25,30,20,33);(1,2,3,4)(5,7)(6,12)(8,25)(9,19)(10,21)(11,23)(13,32)(14,33)(15,34)(16,31)(17,35)(18,29)(20,28)(22,27)(24,30)(26,36);(6,13)(7,19)(8,26)(9,28)(10,27)(12,29)(14,16)(15,17)(18,24)(20,21)(22,36)(23,25)(30,33)(32,34)Polycyclic presentation:
2^9|a^2=g,b^2=d,e^2=h,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fh,h^a=hi,c^b=cf,e^b=eh,f^b=fhi,h^b=hi,e^c=ehi,f^c=fhi,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250420, #72250421, #72250435, … |
| maximal normal subgroup of | #72250420, #72250421, #72250435, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55292);