Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573485 |
| 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×#127 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#127 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1880 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,11)(8,12)(9,10)(13,14,20,19)(15,17,21,27)(16,18,22,28)(23,26,25,24);(6,10)(11,12)(13,15,20,21)(14,17,19,27)(16,24,22,26)(18,23,28,25);(1,2,3,4)(5,7,9,8)(6,11,10,12)(13,16)(14,18)(15,23)(17,26)(19,28)(20,22)(21,25)(24,27);(5,7,9,8)(6,11,10,12)(13,17,20,27)(14,21,19,15)(16,25,22,23)(18,24,28,26);(5,8,9,7)(6,12,10,11)(13,18,20,28)(14,22,19,16)(15,24,21,26)(17,23,27,25)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fg,d^2=fi,e^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6492766);