Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20671 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#117 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#381 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1846 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,15,17,11)(8,19,21,34)(9,14)(10,13)(12,20,25,35)(16,29)(18,28,32,24)(22,26)(23,31)(27,36)(30,33);(1,2,3,4)(5,7)(6,11)(8,16)(9,18)(10,17)(12,23)(13,15)(14,24)(19,33)(20,31)(21,30)(22,32)(25,27)(26,28)(29,34)(35,36);(5,8)(6,12,13,25)(7,16)(9,20)(10,21)(11,23,15,27)(14,19,26,34)(17,30)(18,31)(22,35)(24,33,28,29)(32,36);(7,17)(11,15)(16,30)(18,32)(23,27)(24,28)(29,33)(31,36)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,f^2=i,b^a=be,c^a=cf,e^a=ei,d^b=di,g^b=gi,g^c=gi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72210945, #72210949, #72210958 |
| maximal normal subgroup of | #72210945, #72210949, #72210958 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 260720);