Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437521 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3511 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3536 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)(7,10)(8,9)(11,12)(13,14,19,17)(15,21,25,22)(16,18,27,20)(23,28,26,24);(1,2,3,4)(13,15)(14,21)(16,26)(17,22)(18,24)(19,25)(20,28)(23,27);(6,9)(10,12)(13,16)(14,20)(15,22)(17,18)(19,27)(21,25)(23,24)(26,28);(5,7,8,11)(6,10,9,12)(13,17,19,14)(15,23,25,26)(16,18,27,20)(21,28,22,24);(13,18,19,20)(14,16,17,27)(15,24,25,28)(21,26,22,23)Polycyclic presentation:
2^9|a^2=i,b^2=f,d^2=g,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,h^b=hi,h^c=hi,h^d=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6356243);