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 | #61393 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C8×Q8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
635 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,6,7,4,3,5,8)(9,10,13,18,22,15,24,20,14,17,23,16,12,11,19,21);(5,6)(7,8)(9,11)(10,12)(13,18)(14,15)(16,23)(17,22)(19,20)(21,24);(1,3,4,2)(5,7,6,8)(9,12,14,22)(10,15,17,11)(13,19,23,24)(16,21,18,20)Polycyclic presentation:
2^9|a^2=f,c^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 | #72309476, #72309501, #72309502, … |
| maximal normal subgroup of | #72309476, #72309501, #72309502, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 39098);