Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C2×C22⋊C4 |
| commutator | C23 | C2×C4×C8 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | C2×#336 |
| prime radical | C25 | C2×C8 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
754 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(9,10,12,18,14,11,16,21,15,17,13,19,23,20,24,22);(1,3)(2,5)(4,8)(6,7)(9,11,13,22,14,17,24,18,15,20,12,21,23,10,16,19);(10,17)(11,20)(18,19)(21,22)Polycyclic presentation:
2^9|a^2=e,b^2=f,e^2=h,f^2=h,h^2=i,b^a=bd,c^a=ci,d^a=dg,c^b=ci,e^b=eg128 conjugacy classes — view the full character table.
| maximal subgroup of | #72279467, #72279477, #72279502, … |
| maximal normal subgroup of | #72279467, #72279477, #72279502, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34248);