Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #512862 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2196 |
| commutator, Frattini, cosocle | #127 | C24 |
| LCS | G ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×#127 ◁ G | |
| LECS | G ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
1260 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 1260 conjugacy classes of subgroups — too many to draw. Only its 112 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 4 generators on 16 points:
(4,12)(6,13)(14,15);(1,2)(3,6)(4,8)(5,7)(9,13)(10,15)(11,14)(12,16);(1,3)(2,6)(4,9)(5,11)(7,14)(8,13)(10,12)(15,16);(2,7)(6,14)(8,16)(13,15)Polycyclic presentation:
2^9|g^2=i,b^a=be,c^a=cf,g^a=gh,d^b=di,f^b=fghi,h^b=hi,e^c=eg,g^c=gi,h^c=hi,f^e=fhi,g^e=gi,g^f=gi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72213671, #72216600, #72216601, … |
| maximal normal subgroup of | #72213671, #72216600, #72216601, … |
| maximal quotient of | #72231241, #72231245, #72231248, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420058);