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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2215 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)(7,12)(8,10)(9,11)(13,14,19,23)(15,20,25,18)(16,21,26,28)(17,22,24,27);(5,7,9,8)(6,10,11,12)(13,15,19,25)(14,20,23,18)(16,21,26,28)(17,27,24,22);(1,2,3,4)(13,16,19,26)(14,21,23,28)(15,24,25,17)(18,22,20,27);(5,8,9,7)(6,10,11,12)(15,25)(16,22)(17,28)(18,20)(21,24)(26,27);(13,17,19,24)(14,22,23,27)(15,26,25,16)(18,21,20,28)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=gi,d^2=f,e^2=i,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.
| maximal subgroup of | #72284872 |
| maximal normal subgroup of | #72284872 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6491675);