Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #7519 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1402 |
| commutator | C4⋊Q8 | C22×C4 |
| Frattini, cosocle | #492 | C23 |
| socle | C2 | C2×#1402 |
| prime radical | #18510 | C2 |
| LCS | G ▷ C4⋊Q8 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ #3147 ◁ G | |
| LECS | G ▷ #492 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
340 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 340 conjugacy classes of subgroups — too many to draw. Only its 48 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 — 3 generators on 32 points:
(1,2)(3,8)(4,15,17,7)(5,11)(6,10,18,25)(9,16,21,19)(12,26,28,20)(13,24)(14,23,29,32)(22,27,31,30);(2,7)(4,16)(8,20)(9,11)(10,15)(12,27)(17,19)(21,25)(22,24)(23,26)(28,30)(31,32);(1,3,5,13)(2,8,11,24)(4,12,16,27)(6,14,18,29)(7,20,9,22)(10,23,25,32)(15,26,21,31)(17,28,19,30)Polycyclic presentation:
2^9|a^2=e,c^2=i,d^2=i,f^2=hi,g^2=i,b^a=bd,d^a=df,f^a=fg,g^a=gh,h^a=hi,d^b=di,e^b=efh,f^b=fhi,g^b=ghi,e^d=egh,f^d=fhi,g^d=gi,f^e=fhi,g^e=gi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72283315, #72283352, #72283393, … |
| maximal normal subgroup of | #72283315, #72283352, #72283393, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60643);