Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13472 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2067 |
| 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 | |
722 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)(5,6,10,14,11,15,9,26,12,17,27,32,8,24,25,16)(7,18,22,33,23,31,21,36,19,34,35,30,20,13,28,29);(1,2,3,4)(5,7,8,20,12,19,11,23)(6,13,15,18,17,31,24,34)(9,21,10,22,25,28,27,35)(14,29,26,33,32,36,16,30);(6,14)(7,19)(8,11)(9,27)(10,25)(13,29)(15,16)(17,32)(18,30)(21,22)(24,26)(28,35)(31,36)(33,34)Polycyclic presentation:
2^9|a^2=g,b^2=d,d^2=i,e^2=hi,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fi,c^b=cf,f^b=fi,d^c=di,e^c=eh,g^c=ghi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250135, #72250143, #72250154, … |
| maximal normal subgroup of | #72250135, #72250143, #72250154, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55427);