Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24986 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #555 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2572 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1206 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6,10,15,11,16,9,14)(7,17,21,29,19,28,20,12)(8,22,25,13,26,31,24,32)(18,27,23,30,33,36,34,35);(1,2,4,3)(5,7,11,19)(6,12)(8,23)(9,20,10,21)(13,30,32,35)(14,28)(15,17)(16,29)(18,24)(22,27,31,36)(25,33)(26,34);(5,8)(6,13)(7,18)(9,24)(10,25)(11,26)(12,27)(14,22)(15,31)(16,32)(17,30)(19,33)(20,34)(21,23)(28,35)(29,36);(1,3,4,2)(6,14,16,15)(7,19)(12,17,29,28)(13,22,32,31)(18,33)(20,21)(23,34)(27,30,36,35)Polycyclic presentation:
2^9|a^2=h,b^2=eg,d^2=eh,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,d^b=di,f^c=fi,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#378815, #72216206, #72216224, … |
| maximal normal subgroup of | C2×#378815, #72216206, #72216224, … |
| maximal quotient of | #72215774, #72215776, #72216088, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 286011);