Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518964 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3264 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2281 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(5,6)(7,11)(8,9)(10,12)(13,14)(15,23)(16,27)(17,25)(18,21)(19,20)(22,24)(26,28);(2,3)(5,6)(7,10)(8,9)(11,12)(13,15,24,27,18,26,17,20)(14,19,25,28,21,16,22,23);(1,2,4,3)(6,9)(10,11)(13,16)(14,20)(15,25)(17,28)(18,19)(21,27)(22,26)(23,24);(1,3,4,2)(5,7,8,12)(6,10,9,11)(14,21)(16,19)(22,25)(23,28);(13,15,24,27,18,26,17,20)(14,19,25,28,21,16,22,23)Polycyclic presentation:
2^9|b^2=fi,c^2=h,d^2=gi,e^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di,e^c=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72257096, #72257111, #72257158, … |
| maximal normal subgroup of | #72257096, #72257111, #72257158, … |
| maximal quotient of | #72308896 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7504143);