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 |
| 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 | |
1065 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 1065 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)(3,7)(4,8)(5,17)(6,9)(10,18)(11,25)(12,21)(13,20)(14,23)(15,24)(16,22)(19,26)(27,31)(28,32)(29,30);(1,3,6,13)(2,7,9,20)(4,11,16,28)(5,12,18,29)(8,19,22,31)(10,21,17,30)(14,26,15,27)(23,25,24,32);(1,4)(2,8)(3,11)(5,14)(6,16)(7,19)(9,22)(10,23)(12,26)(13,28)(15,18)(17,24)(20,31)(21,25)(27,29)(30,32);(2,9)(7,20)(8,22)(10,17)(19,31)(21,30)(23,24)(25,32)Polycyclic presentation:
2^9|b^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 | #72281670, #72306592, #72306594, … |
| maximal normal subgroup of | #72281670, #72306592, #72306594, … |
| maximal quotient of | #72282068 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420055);