Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3585 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C42 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C4 | C23×C42 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3304 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,3)(9,10,13,16)(11,15,18,22)(12,14,20,17)(19,21,24,23);(5,6,8,7)(10,14)(12,20)(15,21)(16,17)(19,24)(22,23);(5,7,8,6)(9,11)(10,15)(12,19)(13,18)(14,21)(16,22)(17,23)(20,24);(1,3,4,2)(10,16)(14,17)(15,22)(21,23);(11,18)(15,22)(19,24)(21,23)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=h,d^2=gi,f^2=i,b^a=bf,d^a=di,f^a=fi,f^b=fi,e^c=ei164 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#60176, #72257432, #72257730, … |
| maximal normal subgroup of | C4×#60176, #72257432, #72257730, … |
| maximal quotient of | #72265244, #72280295 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6277270);