Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8×#120 |
| 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×#121 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2037 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6)(7,11)(8,12)(9,10);(6,10)(11,12)(13,14,19,22)(15,23,26,28)(16,20,27,25)(17,21,18,24);(2,4)(13,15,19,26)(14,16,22,27)(17,25,18,20)(21,28,24,23);(5,7,9,8)(6,11,10,12)(13,16)(14,20)(15,24)(17,23)(18,28)(19,27)(21,26)(22,25);(5,8,9,7)(6,12,10,11)(13,17)(15,25)(18,19)(20,26)(21,24)(23,28)Polycyclic presentation:
2^9|b^2=i,c^2=i,d^2=f,e^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6484963);