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 | |
1998 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)(7,17)(8,13)(9,14)(10,15)(11,16)(12,19)(18,32)(20,26)(21,27)(22,28)(23,30)(24,29)(25,34)(31,35)(33,36);(5,7,10,21,11,20,9,19)(6,12,14,26,16,27,15,17)(8,18,23,34,24,35,22,33)(13,25,28,32,29,36,30,31);(1,3,4,2)(5,8)(6,13)(7,18)(9,22)(10,23)(11,24)(12,25)(14,28)(15,30)(16,29)(17,31)(19,33)(20,35)(21,34)(26,32)(27,36);(5,9,11,10)(6,14,16,15)(7,19,20,21)(8,22,24,23)(12,26,27,17)(13,28,29,30)(18,33,35,34)(25,32,36,31)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=g,d^2=i,e^2=i,h^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, #72210987, #72313078, … |
| maximal normal subgroup of | #72210949, #72210987, #72313078, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261100);