Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #148483 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #11174 |
| commutator | C22×D4 | C22×C4 |
| Frattini, cosocle | C23×D4 | C23 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C22 ▷ C2 ▷ C1 | |
880 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 880 conjugacy classes of subgroups — too many to draw. Only its 120 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)(3,12,16,24)(4,17,21,31)(5,22,6,10)(7,11)(8,14)(9,20)(13,28)(15,25)(18,26,29,32)(19,27)(23,30);(1,3)(2,8)(4,18)(5,15)(6,14)(7,16)(9,26)(10,24)(11,25)(12,22)(13,20)(17,23)(19,30)(21,29)(27,32)(28,31);(1,4,5,19,7,21,6,20)(2,9,10,17,11,27,22,31)(3,13,14,29,16,30,15,18)(8,23,12,26,25,28,24,32)Polycyclic presentation:
2^9|a^2=g,c^2=d,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fhi,h^a=hi,c^b=cf,d^b=di,e^b=ehi,f^b=fi,g^b=ghi,e^c=eh,g^c=gh,h^c=hi,e^d=ei,g^d=gi,f^e=fi,g^e=gi,g^f=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72256012, #72262398, #72262402 |
| maximal normal subgroup of | #72256012, #72262398, #72262402 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55679);