Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #512884 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2196 |
| commutator, Frattini, cosocle | #124 | C24 |
| prime radical | #61110 | C2 |
| LCS | G ▷ #124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G | |
| LECS | G ▷ #124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
1037 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 1037 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) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,13)(3,14,19,32)(4,12,21,9)(5,11,24,10)(7,22,27,16)(8,28,25,17)(15,29,18,26)(20,30,23,31);(1,3)(2,7)(4,18)(5,22)(6,19)(8,9)(10,32)(11,14)(12,25)(13,27)(15,21)(16,24)(17,20)(23,28)(26,30)(29,31);(1,3)(2,8)(4,15)(5,17)(6,19)(7,9)(10,32)(11,14)(12,27)(13,25)(16,23)(18,21)(20,22)(24,28)(26,31)(29,30);(1,4,6,21)(2,9,13,12)(3,15,19,18)(5,20,24,23)(7,25,27,8)(10,30,11,31)(14,26,32,29)(16,17,22,28)Polycyclic presentation:
2^9|a^2=i,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,g^a=gh,d^b=di,e^b=ei,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 | #72337616, #72337673, #72340538 |
| maximal normal subgroup of | #72337616, #72337673, #72340538 |
| maximal quotient of | #72337620, #72337677 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420082);