Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1909 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#386 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2030 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6)(7,16)(8,13)(9,14)(10,23)(11,15)(12,19)(17,32)(18,26)(20,31)(21,27)(22,29)(24,28)(25,33)(30,34)(35,36);(5,7)(6,12,15,26)(8,17)(9,18)(10,19)(11,20)(13,25,29,36)(14,16,23,31)(21,33)(22,34)(24,35)(27,30,28,32);(1,3,4,2)(5,8)(6,13)(7,17)(9,21)(10,24)(11,22)(12,25)(14,27)(15,29)(16,30)(18,33)(19,35)(20,34)(23,28)(26,36)(31,32);(5,9,11,10)(6,14,15,23)(7,18,20,19)(8,21,22,24)(12,16,26,31)(13,27,29,28)(17,33,34,35)(25,30,36,32)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=g,d^2=i,e^2=i,b^a=be,c^a=cf,e^a=ei,h^a=hi,f^b=fi,e^c=ei116 conjugacy classes — view the full character table.
| maximal subgroup of | #72210949, #72210958, #72312515, … |
| maximal normal subgroup of | #72210949, #72210958, #72312515, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261098);