Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #148246 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #11087 |
| commutator | C22×D4 | C22×C4 |
| Frattini, cosocle | C23×D4 | C23 |
| prime radical | C2×#534 | C22 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C22 ▷ C2 ▷ C1 | |
932 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,11)(4,15,18,8)(5,9)(6,10)(7,13)(12,26,16,24)(14,25)(17,32,27,22)(19,23)(20,29,31,28)(21,30);(1,3,5,13)(2,7,9,11)(4,16,17,29)(6,14,19,30)(8,20,22,26)(10,21,23,25)(12,27,28,18)(15,31,32,24);(1,4)(2,8)(3,12)(5,17)(6,18)(7,20)(9,22)(10,15)(11,24)(13,29)(14,16)(19,27)(21,31)(23,32)(25,26)(28,30)Polycyclic presentation:
2^9|a^2=g,b^2=d,b^a=bd,c^a=ce,e^a=ei,f^a=fhi,c^b=cf,f^b=fh,h^b=hi,d^c=dh,g^c=gi,f^d=fi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72238712, #72238717, #72238720, … |
| maximal normal subgroup of | #72238712, #72238717, #72238720, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55442);