Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #153622 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13673 |
| commutator, Frattini, cosocle | #494 | C23 |
| prime radical | #3293 | C22 |
| LCS | G ▷ #494 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ #494 ◁ G | |
| LECS | G ▷ #494 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
258 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 258 conjugacy classes of subgroups — too many to draw. Only its 30 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)(3,9)(4,17)(5,20)(6,24)(7,12)(8,14)(10,23)(11,18)(13,16)(15,29)(19,27)(21,26)(22,30)(25,32)(28,31);(1,3,7,16)(2,8,12,26)(4,18,15,6)(5,21,23,14)(9,10,13,20)(11,28,24,27)(17,25,29,30)(19,22,31,32);(1,4,7,15)(2,9,12,13)(3,6,16,18)(5,22,23,32)(8,20,26,10)(11,29,24,17)(14,31,21,19)(25,27,30,28)Polycyclic presentation:
2^9|b^2=i,c^2=i,d^2=i,e^2=i,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fg,g^a=gi,h^a=hi,c^b=ci,d^b=di,e^b=efgh,g^b=gh,d^c=df,e^c=ei,f^c=fhi,g^c=ghi,e^d=egh,f^d=fhi,g^d=gi,f^e=fhi,g^e=gi35 conjugacy classes — view the full character table.
| maximal subgroup of | #72283477 |
| maximal normal subgroup of | #72283477 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60820);