Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #145380 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #10473 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#2213 | C2 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
325 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 325 conjugacy classes of subgroups — too many to draw. Only its 74 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:
(2,7)(3,11)(4,16)(5,18)(8,12)(9,25)(10,21)(13,14)(15,29)(17,20)(19,32)(22,24)(23,26)(30,31);(1,2)(3,12)(4,17)(5,20)(6,10)(7,16)(8,14)(9,18)(11,26)(13,24)(15,22)(19,25)(21,32)(23,27)(28,30)(29,31);(1,3,4,14,18,29,32,27,6,15,19,13,5,11,16,28)(2,8,21,22,20,30,17,31,10,24,7,12,9,23,25,26)Polycyclic presentation:
2^9|c^2=dh,d^2=gi,e^2=h,f^2=gi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fg,g^a=gi,c^b=cf,d^b=dg,e^b=egi,f^b=fg,g^b=gi,h^b=hi,d^c=di,e^c=egh,h^c=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 52576);