Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23387 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2074 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(9,10,11,12)(14,18)(15,22)(16,19)(20,23)(21,26)(24,27);(5,6)(7,8)(13,14,19,26,17,21,16,18)(15,23,27,25,24,20,22,28);(2,4)(6,8)(13,15,16,22,17,24,19,27)(14,20,18,25,21,23,26,28);(15,24)(20,23)(22,27)(25,28)Polycyclic presentation:
2^9|a^2=h,b^2=ei,c^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,c^b=cg,d^c=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72315759, #72315780, #72315825, … |
| maximal normal subgroup of | #72315759, #72315780, #72315825, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265234);