Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #153448 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#1402 |
| commutator | C4⋊Q8 | C22×C4 |
| Frattini, cosocle | #492 | C23 |
| prime radical | #18684 | C2 |
| LCS | G ▷ C4⋊Q8 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×C42 ◁ C2×#492 ◁ G | |
| LECS | G ▷ #492 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
330 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 330 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)(3,11)(4,15,17,26)(5,10)(6,9,19,27)(7,13,22,30)(8,18,25,20)(12,24)(14,28,29,32)(16,23,31,21);(1,3)(2,7)(4,16)(5,12)(6,14)(8,24)(9,22)(10,21)(11,26)(13,18)(15,28)(17,31)(19,29)(20,30)(23,27)(25,32);(3,12)(7,21)(11,24)(13,16)(14,29)(22,23)(28,32)(30,31)Polycyclic presentation:
2^9|a^2=e,d^2=i,f^2=hi,g^2=i,b^a=bd,d^a=df,f^a=fg,g^a=gh,h^a=hi,c^b=ci,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=gi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72282994, #72283007, #72283015, … |
| maximal normal subgroup of | #72282994, #72283007, #72283015, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60644);