Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62243 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
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 | #60940 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2036 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)(3,5)(4,7)(6,8)(9,10,13,17,14,11,18,16)(12,15,21,23,20,19,24,22);(2,5)(3,8)(6,7)(9,11)(10,14)(12,19)(13,16)(15,20)(17,18)(21,22)(23,24);(9,12,13,21,14,20,18,24)(10,15,17,23,11,19,16,22);(12,20)(15,19)(21,24)(22,23);(10,11)(15,19)(16,17)(22,23)Polycyclic presentation:
2^9|a^2=g,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 | #72287782, #72319647, #72319808 |
| maximal normal subgroup of | #72287782, #72319647, #72319808 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6252801);