Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #131812 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1786 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #61234 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
606 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,3)(4,5)(7,8)(9,10,13,17,16,21,23,24,14,18,12,22,15,11,19,20);(1,2,6,8)(3,4,7,5)(9,11,14,21)(10,16,18,15)(12,22,13,17)(19,24,23,20);(1,3)(2,4)(5,8)(6,7)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=hi,e^2=gi,f^2=gh,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,d^b=dh,e^b=egi,f^b=fgh,h^b=hi,d^c=dgi,e^c=egi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72309556, #72309577, #72309578, … |
| maximal normal subgroup of | #72309556, #72309577, #72309578, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 39128);