Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62892 |
| 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×#3522 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7008 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2,4,3)(6,8)(10,12)(13,14)(15,17)(16,24)(18,22)(19,20)(21,27)(23,26)(25,28);(1,3,4,2)(14,20)(15,22)(21,24)(23,28);(5,6,9,8)(7,10,11,12)(13,15,19,22)(14,17,20,18)(16,23,27,28)(21,25,24,26);(7,11)(10,12)(13,16,19,27)(14,21,20,24)(15,23,22,28)(17,25,18,26);(5,7)(6,10)(8,12)(9,11)(13,17,19,18)(14,15,20,22)(16,25,27,26)(21,23,24,28);(5,8,9,6)(7,12,11,10)(13,18,19,17)(14,22,20,15)(16,26,27,25)(21,28,24,23)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=gh,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=cg,e^d=eh146 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7533955);