Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #148270 |
| 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 | |
920 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)(5,19)(6,10)(7,13)(9,20)(12,26,16,24)(14,25)(17,23,27,32)(21,29,31,28)(22,30);(1,3,5,13,6,14,20,30)(2,7,9,11,10,22,19,25)(4,16,17,29,18,12,27,28)(8,21,23,26,15,31,32,24);(1,4)(2,8)(3,12)(5,17)(6,18)(7,21)(9,23)(10,15)(11,24)(13,29)(14,16)(19,32)(20,27)(22,31)(25,26)(28,30)Polycyclic presentation:
2^9|a^2=g,b^2=d,d^2=i,b^a=bd,c^a=ce,d^a=di,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 | #72255811, #72256132, #72262398, … |
| maximal normal subgroup of | #72255811, #72256132, #72262398, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55466);