Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152123 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #12697 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C2×C4⋊Q8 | C23 |
| prime radical | #3498 | C4 |
| LCS | G ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ #117 ◁ #3509 ◁ G | |
| LECS | G ▷ C2×C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
653 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 653 conjugacy classes of subgroups — too many to draw. Only its 76 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:
(1,2,5,11,7,13,21,29)(3,9,14,26,4,16,17,31)(6,12,22,28,23,10,20,27)(8,19,25,18,24,30,32,15);(1,3)(2,8)(4,7)(5,19)(6,18)(9,12)(10,16)(11,26)(13,24)(14,22)(15,23)(17,20)(21,30)(25,27)(28,32)(29,31);(1,4,7,3)(2,9)(5,17,21,14)(6,18,23,15)(8,12)(10,24)(11,26)(13,16)(19,22,30,20)(25,28)(27,32)(29,31)Polycyclic presentation:
2^9|a^2=f,c^2=h,d^2=hi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,c^b=ch,d^b=dhi,e^b=ei,f^b=fgh,g^b=gi,e^c=ei,f^d=fhi,g^f=gi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72340147, #72340149, #72340160, … |
| maximal normal subgroup of | #72340147, #72340149, #72340160, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59321);