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 | #60646 | C2 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×Q8 ◁ C2×#41 ◁ #3499 ◁ G | |
| LECS | G ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
1103 conjugacy classes of subgroups; only the 192 normal subgroups are shown.
This group has 1103 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:
(1,2,6,12)(3,15,17,28)(4,10,19,26)(5,11,20,27)(7,13,23,29)(8,14,22,30)(9,21,24,16)(18,25,31,32);(1,3,8,4)(2,9,14,25)(5,16,7,18)(6,21,22,31)(10,13,15,11)(12,28,30,26)(17,20,19,23)(24,29,32,27);(1,4,8,3)(2,10,14,15)(5,16,7,18)(6,19,22,17)(9,13,25,11)(12,26,30,28)(20,21,23,31)(24,29,32,27);(1,5,8,7)(2,11,14,13)(3,16,4,18)(6,20,22,23)(9,10,25,15)(12,27,30,29)(17,21,19,31)(24,26,32,28)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,d^2=i,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 | #72338304, #72338305, #72338308, … |
| maximal normal subgroup of | #72338304, #72338305, #72338308, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419812);