Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573615 |
| 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×#496 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1832 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)(13,14,17,16)(15,21,23,28)(18,25,26,20)(19,27,22,24);(1,2,3,4)(6,10)(8,11)(13,14)(15,22)(16,17)(18,20)(19,23)(21,27)(24,28)(25,26);(5,6)(7,8)(9,10)(11,12)(13,15,17,23)(14,19,16,22)(18,27,26,24)(20,28,25,21);(5,7,9,12)(6,11,10,8)(13,16,17,14)(15,19,23,22)(18,25,26,20)(21,27,28,24);(5,8,9,11)(6,7,10,12)(14,16)(19,22)(20,25)(21,28)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=i,d^2=h,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6525073);