Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23391 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
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 | C22×#127 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2654 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(9,10,12,11)(13,14)(15,18)(16,19)(17,20);(2,4)(5,6)(7,8)(14,18)(15,16)(19,20);(6,8)(14,19)(15,16)(18,20);(9,11,12,10)(13,15,17,16)(14,18,20,19)Polycyclic presentation:
2^9|a^2=h,d^2=hi,e^2=i,f^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,c^b=cg,d^b=di,e^b=ei,f^b=fi,d^c=di,e^c=ei,f^c=fi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72217689, #72217728, #72264377, … |
| maximal normal subgroup of | #72217689, #72217728, #72264377, … |
| maximal quotient of | #72216608, #72216614 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266239);