Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #187878 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | #62840 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2151 conjugacy classes of subgroups; only the 310 normal subgroups are shown.
This group has 2151 conjugacy classes of subgroups — too many to draw. Only its 310 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) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,9,6,11,5,10)(3,8,12,15,13,4,14,16)(17,18)(19,26)(20,28)(21,31)(22,24)(23,27)(25,32)(29,30);(1,3)(2,8)(4,11)(5,12)(6,13)(7,14)(9,16)(10,15)(17,19)(18,23)(20,24)(21,28)(22,29)(25,30)(26,31)(27,32);(1,4,6,8)(2,3,11,13)(5,16,7,15)(9,14,10,12)(17,20,22,23)(18,19,24,29)(21,30,32,26)(25,27,31,28);(1,5,6,7)(2,9,11,10)(3,12,13,14)(4,15,8,16)(18,24)(20,23)(25,31)(26,30)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=hi,e^2=i,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,e^b=ei,f^b=fhi,g^b=ghi,d^c=dh,e^c=ehi,f^c=fh,g^c=ghi80 conjugacy classes — view the full character table.
| Fitting subgroup of | #72272607 |
| maximal subgroup of | #72240112, #72272564, #72272568, … |
| maximal normal subgroup of | #72240112, #72272564, #72272568, … |
| maximal quotient of | #72341053 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 95118);