Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #512876 |
| 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 |
| LCS | G ▷ #124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×#124 ◁ G | |
| LECS | G ▷ #124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
1053 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 1053 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)(3,13)(4,11)(5,12)(6,9)(7,14)(8,15)(10,20)(16,26)(17,29)(18,25)(19,31)(21,28)(22,30)(23,24)(27,32);(1,3)(2,7)(4,18)(5,16)(6,17)(8,24)(9,25)(10,29)(11,13)(12,27)(14,20)(15,22)(19,32)(21,23)(26,30)(28,31);(1,3)(2,8)(4,15)(5,21)(6,17)(7,24)(9,27)(10,29)(11,13)(12,25)(14,19)(16,23)(18,22)(20,32)(26,31)(28,30);(2,9)(7,25)(8,27)(10,11)(12,24)(13,29)(26,28)(30,31)Polycyclic presentation:
2^9|e^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,g^a=gh,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=gi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72279927, #72286052, #72340538 |
| maximal normal subgroup of | #72279927, #72286052, #72340538 |
| maximal quotient of | #72340618, #72340620, #72340679, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420072);