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×#576 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ #2965 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1824 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)(9,10,13,15)(11,16,18,22)(12,14,20,17)(19,21,24,23);(10,14)(12,20)(15,17)(16,21)(19,24)(22,23);(1,3,4,8,5,2,6,7)(10,15)(14,17)(16,22)(21,23);(2,3)(7,8)(9,11)(10,16)(12,19)(13,18)(14,21)(15,22)(17,23)(20,24);(10,15)(11,18)(14,17)(19,24)Polycyclic presentation:
2^9|a^2=g,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.
| maximal subgroup of | #72322950, #72343921 |
| maximal normal subgroup of | #72322950, #72343921 |
| maximal quotient of | #72343939 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6252300);