Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #512858 |
| 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 | #127 | C24 |
| prime radical | #61105 | C2 |
| LCS | G ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G | |
| LECS | G ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
1024 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 1024 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:
(1,2,8,13)(3,14,18,20)(4,17,22,16)(5,12,24,9)(6,10,23,30)(7,11,26,28)(15,31,32,25)(19,27,21,29);(1,3)(2,4)(5,15)(6,16)(7,27)(8,18)(9,19)(10,14)(11,31)(12,21)(13,22)(17,23)(20,30)(24,32)(25,28)(26,29);(1,4)(2,3)(5,19)(6,25)(7,20)(8,22)(9,15)(10,29)(11,17)(12,32)(13,18)(14,26)(16,28)(21,24)(23,31)(27,30);(1,5,8,24)(2,9,13,12)(3,15,18,32)(4,19,22,21)(6,7,23,26)(10,28,30,11)(14,25,20,31)(16,27,17,29)Polycyclic presentation:
2^9|a^2=i,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,g^a=gh,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 | #72283528, #72283572, #72283670, … |
| maximal normal subgroup of | #72283528, #72283572, #72283670, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420057);