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×#3179 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1827 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,9,7)(8,11,12,10)(14,19)(16,25)(17,26)(20,27)(21,28)(22,24);(1,2)(3,4)(5,7,9,6)(8,10,12,11)(13,14,17,19,18,21,26,28)(15,20,24,27,23,16,22,25);(5,8,9,12)(6,10,7,11)(13,15)(14,20)(16,21)(17,24)(18,23)(19,27)(22,26)(25,28);(2,4)(6,7)(10,11)(14,21)(15,23)(19,28)(22,24);(13,16,26,27,18,20,17,25)(14,22,28,23,21,24,19,15)Polycyclic presentation:
2^9|a^2=g,b^2=fg,c^2=g,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.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7453033);