Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #333271 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#1625 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
838 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6,9,10,8,12,7,11)(13,14)(15,21)(16,19)(17,20)(18,23)(22,27)(24,26)(25,28);(5,7)(8,9)(10,11)(13,15)(14,18)(16,23)(17,22)(19,21)(20,26)(24,25)(27,28);(2,4)(5,7,8,9)(6,10,12,11)(15,22)(17,25)(18,26)(20,28)(21,27)(23,24);(2,4)(5,6)(7,10)(8,12)(9,11)(14,19)(15,23)(20,28)(22,24)Polycyclic presentation:
2^9|a^2=ehi,c^2=h,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,e^b=eh,g^b=gi,g^c=gi,e^d=eh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 240487);