Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2214 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)(14,20)(15,23)(16,18)(17,25)(21,28)(22,24);(2,3)(5,6)(7,10)(8,9)(11,12)(13,14,18,20,19,24,16,22)(15,25,26,28,23,21,27,17);(1,2,4,3)(6,9)(10,11)(13,15)(14,21)(16,27)(17,20)(18,26)(19,23)(22,28)(24,25);(1,3,4,2)(5,7,8,12)(6,10,9,11)(13,16,19,18)(14,22,24,20)(15,26,23,27)(17,25,28,21);(13,17,18,25,19,28,16,21)(14,23,20,27,24,15,22,26)Polycyclic presentation:
2^9|b^2=f,c^2=h,d^2=g,e^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^b=ei,d^c=di,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7504213);