Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #63041 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C26 |
| commutator | C22 | C25×C4 |
| prime radical | C2×#3560 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7082 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2,4,3)(5,6)(7,11)(8,9)(10,12)(13,14)(15,22)(16,18)(17,19)(20,25)(21,26)(23,27)(24,28);(6,9)(10,11)(13,15)(14,20)(16,23)(17,22)(18,24)(19,25)(21,27)(26,28);(5,6)(7,10)(8,9)(11,12)(13,16,19,26)(14,21,17,18)(15,23,25,28)(20,27,22,24);(5,7,8,12)(6,10,9,11)(13,17,19,14)(15,20,25,22)(16,18,26,21)(23,27,28,24);(1,3,4,2)(13,18)(14,16)(15,24)(17,26)(19,21)(20,23)(22,28)(25,27);(13,15)(14,22)(16,23)(17,20)(18,27)(19,25)(21,24)(26,28)Polycyclic presentation:
2^9|a^2=i,c^2=h,d^2=g,e^2=i,b^a=bg,c^a=ch,d^a=dg,c^b=cgh,d^b=dh,f^e=fh140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7534706);