Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62243 |
| 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×#532 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1812 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,5)(3,7)(6,8)(9,10,14,17,15,11,13,19)(12,16,22,23,20,18,21,24);(1,2,4,6)(3,8,7,5)(9,11,15,10)(12,18,20,16)(13,17,14,19)(21,23,22,24);(9,12,14,22,15,20,13,21)(10,16,17,23,11,18,19,24);(12,20)(16,18)(21,22)(23,24);(10,11)(16,18)(17,19)(23,24)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=g,f^2=h,g^2=i,b^a=bf,e^a=ei,f^a=fh,e^b=ei,f^b=fh,d^c=di128 conjugacy classes — view the full character table.
| maximal subgroup of | #72318537, #72319647, #72319808 |
| maximal normal subgroup of | #72318537, #72319647, #72319808 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6252802);