Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #512862 |
| 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 | #61110 | C2 |
| 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 | |
1066 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 1066 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:
(1,2,6,10)(3,7,12,20)(4,8,16,24)(5,17,18,9)(11,25,26,32)(13,21,29,30)(14,22,15,23)(19,28,31,27);(1,3)(2,7)(4,11)(5,13)(6,12)(8,19)(9,21)(10,20)(14,27)(15,28)(16,26)(17,30)(18,29)(22,25)(23,32)(24,31);(1,4)(2,8)(3,11)(5,14)(6,16)(7,19)(9,22)(10,24)(12,26)(13,27)(15,18)(17,23)(20,31)(21,25)(28,29)(30,32);(3,12)(7,20)(11,26)(13,29)(19,31)(21,30)(25,32)(27,28)Polycyclic presentation:
2^9|a^2=i,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 | #72231244, #72231248, #72231250, … |
| maximal normal subgroup of | #72231244, #72231248, #72231250, … |
| maximal quotient of | C2×#512863, #72279353, #72279359, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420059);