Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #493657 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #20609 |
| commutator | C2×Q8 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| prime radical | #63037 | C2 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×2−1+4 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
2187 conjugacy classes of subgroups; only the 304 normal subgroups are shown.
This group has 2187 conjugacy classes of subgroups — too many to draw. Only its 304 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 — 4 generators on 32 points:
(1,2)(3,15)(4,9,17,24)(5,11,22,28)(6,12,18,10)(7,13)(8,14)(16,29,21,27)(19,23)(20,26)(25,31)(30,32);(1,3,8,7)(2,9,14,24)(4,18,17,6)(5,16,22,21)(10,15,12,13)(11,23,28,32)(19,31,30,20)(25,29,26,27);(1,4,8,17)(2,10,14,12)(3,6,7,18)(5,19,22,30)(9,13,24,15)(11,25,28,26)(16,20,21,31)(23,27,32,29);(1,5)(2,11)(3,16)(4,19)(6,20)(7,21)(8,22)(9,23)(10,25)(12,26)(13,27)(14,28)(15,29)(17,30)(18,31)(24,32)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,h^a=hi,c^b=ci,e^b=ei,f^b=fhi,g^b=ghi,e^c=eh,f^c=fi,g^c=gi,g^d=gi,f^e=fi,g^e=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400855);