Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #512613 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×#353 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | #127 | C24 |
| prime radical | #60647 | C2 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×D4 ◁ C2×#41 ◁ #3498 ◁ G | |
| LECS | G ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
1153 conjugacy classes of subgroups; only the 192 normal subgroups are shown.
This group has 1153 conjugacy classes of subgroups — too many to draw. Only its 192 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)(3,12,14,7)(4,8)(5,10,19,9)(6,11)(13,26,28,20)(15,21,18,22)(16,23)(17,25,29,24)(27,32,30,31);(1,3)(2,7)(4,13)(5,18)(6,15)(8,20)(9,21)(10,12)(11,22)(14,19)(16,27)(17,30)(23,31)(24,32)(25,26)(28,29);(1,4)(2,8)(3,13)(5,17)(6,16)(7,20)(9,24)(10,25)(11,23)(12,26)(14,28)(15,27)(18,30)(19,29)(21,32)(22,31);(4,16)(8,23)(13,27)(17,29)(20,31)(24,25)(26,32)(28,30)Polycyclic presentation:
2^9|a^2=f,e^2=hi,g^2=i,b^a=be,e^a=eg,g^a=gh,h^a=hi,e^b=ehi,f^b=fgh,g^b=gi,d^c=di,f^e=fhi,g^f=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72317553, #72317638, #72318931, … |
| maximal normal subgroup of | #72317553, #72317638, #72318931, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419809);