Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60454 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C23×C22⋊C4 |
| prime radical | C2×#3449 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5158 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,10,14,11,15,9,24)(7,16,20,26,18,30,19,12)(8,13,23,28,21,27,22,34)(17,29,33,35,31,36,32,25);(5,7)(6,12,15,26)(8,17)(9,19)(10,20)(11,18)(13,25,27,35)(14,16,24,30)(21,31)(22,32)(23,33)(28,29,34,36);(1,2,3,4)(5,8)(6,13)(7,17)(9,22)(10,23)(11,21)(12,25)(14,28)(15,27)(16,29)(18,31)(19,32)(20,33)(24,34)(26,35)(30,36);(8,21)(13,27)(17,31)(22,23)(25,35)(28,34)(29,36)(32,33);(7,18)(12,26)(16,30)(17,31)(19,20)(25,35)(29,36)(32,33)Polycyclic presentation:
2^9|a^2=h,b^2=fh,c^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,e^b=ei,d^c=di116 conjugacy classes — view the full character table.
| maximal subgroup of | #72258207, #72262971, #72262983, … |
| maximal normal subgroup of | #72258207, #72262971, #72262983, … |
| maximal quotient of | #72258212, #72258222, #72258568, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6279159);