Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62253 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C22 | C23×C22⋊C4 |
| prime radical | #62812 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
3936 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)(4,6)(7,8)(9,10,13,17,14,11,18,16)(12,19,21,23,20,15,22,24);(2,5)(3,8)(6,7)(9,11)(10,14)(12,19)(13,16)(15,20)(17,18)(21,23)(22,24);(2,6)(5,7)(9,12)(10,15)(11,19)(13,21)(14,20)(16,23)(17,24)(18,22);(12,20)(15,19)(21,22)(23,24);(10,11)(15,19)(16,17)(23,24)Polycyclic presentation:
2^9|a^2=g,f^2=h,g^2=i,b^a=bf,c^a=ch,e^a=ei,f^a=fh,e^b=ei,f^b=fh,d^c=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72287782, #72319647, #72319800 |
| maximal normal subgroup of | #72287782, #72319647, #72319800 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6250428);