Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #512858 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
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 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G | |
| LECS | G ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
1068 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 1068 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:
(4,15)(5,17)(7,18)(8,22)(11,25)(19,30)(20,21)(23,32)(26,27)(29,31);(1,2)(3,7)(4,8)(5,9)(6,10)(11,19)(12,18)(13,20)(14,21)(15,22)(16,23)(17,24)(25,31)(26,29)(27,30)(28,32);(1,3)(2,7)(4,11)(5,12)(6,14)(8,19)(9,18)(10,21)(13,17)(15,25)(16,26)(20,24)(22,31)(23,29)(27,28)(30,32);(1,4,6,15)(2,8,10,22)(3,11,14,25)(5,16,17,28)(7,19,21,31)(9,23,24,32)(12,26,13,27)(18,29,20,30)Polycyclic presentation:
2^9|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 | #72283303, #72283342, #72283419, … |
| maximal normal subgroup of | #72283303, #72283342, #72283419, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420056);