Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C2×#459 | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ #1744 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
794 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6,10,11,9,8,7,12)(13,14,17,22)(15,23,25,19)(16,21,18,20)(24,28,26,27);(1,2,3,4)(5,7,9,10)(6,11,8,12)(13,15)(14,19)(16,26)(17,25)(18,24)(20,27)(21,28)(22,23);(6,8)(11,12)(14,20)(16,18)(19,27)(21,22)(23,28)(24,26);(5,8,10,12,9,6,7,11)(13,16,17,18)(14,21,22,20)(15,24,25,26)(19,27,23,28)Polycyclic presentation:
2^9|a^2=g,b^2=eh,d^2=g,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,d^b=dhi,d^c=dhi,f^c=fh104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 117764);