Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #512880 |
| 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 | |
1092 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 1092 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:
(2,7)(4,15)(8,24)(9,16)(10,23)(11,27)(12,29)(13,19)(18,20)(22,26)(25,30)(28,32);(1,2)(3,8)(4,16)(5,11)(6,9)(7,15)(10,21)(12,30)(13,26)(14,25)(17,32)(18,27)(19,28)(20,23)(22,31)(24,29);(1,3)(2,8)(4,17)(5,19)(6,14)(7,22)(9,25)(10,26)(11,28)(12,18)(13,21)(15,31)(16,32)(20,29)(23,24)(27,30);(2,9)(7,16)(8,25)(10,11)(22,32)(23,27)(24,30)(26,28)Polycyclic presentation:
2^9|e^2=i,g^2=i,b^a=be,c^a=cf,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=gi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72213674, #72219528, #72219529, … |
| maximal normal subgroup of | #72213674, #72219528, #72219529, … |
| maximal quotient of | #72338280, #72339623 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420076);