Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33856 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#508 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #1772 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
610 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4)(6,13)(7,9)(8,20)(10,11)(14,29)(15,33)(16,28)(17,34)(18,19)(21,32)(22,26)(23,25)(27,31)(30,36);(5,6)(7,13)(8,21)(9,23)(10,15)(11,25)(12,16)(14,17)(18,33)(19,28)(20,36)(22,29)(24,30)(26,27)(31,35)(32,34);(5,7,11,18,12,19,10,9)(6,13,25,33,16,28,15,23)(8,17,26,35,22,34,20,24)(14,27,31,29,32,36,30,21);(5,8)(6,14)(7,17)(9,24)(10,20)(11,26)(12,22)(13,27)(15,30)(16,32)(18,35)(19,34)(21,23)(25,31)(28,36)(29,33)Polycyclic presentation:
2^9|a^2=g,c^2=fi,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,h^a=hi,d^b=de,e^b=eh,h^b=hi,e^d=eh,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72216402, #72216412, #72216413, … |
| maximal normal subgroup of | #72216402, #72216412, #72216413, … |
| maximal quotient of | #72225597 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405087);