Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61071 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | C2×#577 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ #2965 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1770 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,7,5,3,6,8)(9,10)(11,19)(12,18)(13,16)(14,17)(15,20)(21,24)(22,23);(9,11,14,21)(10,15,17,23)(12,20,13,22)(16,19,18,24);(1,3,4,8,5,2,6,7)(9,12,14,13)(10,16,17,18)(11,20,21,22)(15,19,23,24);(2,3)(7,8)(9,10)(11,15)(12,16)(13,18)(14,17)(19,20)(21,23)(22,24);(10,17)(15,23)(16,18)(19,24)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=gh,f^2=h,g^2=i,b^a=bf,c^a=ch,d^a=di,e^a=eh,f^a=fh,f^b=fh,d^c=di,e^d=eh110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6252301);