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×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2020 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)(5,6,8,9)(7,10,11,12);(6,9)(10,12)(13,14,18,22)(15,20,25,27)(16,19,23,28)(17,26,24,21);(2,4)(13,15,18,25)(14,19,22,28)(16,17,23,24)(20,21,27,26);(5,7,8,11)(6,10,9,12)(15,23)(16,25)(17,24)(19,27)(20,28)(21,26);(13,16,18,23)(14,20,22,27)(15,24,25,17)(19,26,28,21)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=ehi,h^b=hi,d^c=dh,e^c=ei,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72307127, #72307195, #72307281 |
| maximal normal subgroup of | #72307127, #72307195, #72307281 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6483347);