Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13436 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2044 |
| commutator | C2×#41 | C22×C4 |
| Frattini, cosocle | C22×#41 | C23 |
| socle | C2 | #11174 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
769 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,8,14)(3,15,10,22)(4,19)(5,16)(6,13)(7,12)(9,27)(11,25)(17,29)(18,23)(20,24,30,26)(21,31,28,32);(1,3)(2,9)(4,20)(5,17)(6,18)(7,19)(8,10)(11,15)(12,21)(13,28)(14,29)(16,22)(23,30)(24,31)(25,27)(26,32);(1,4,5,21)(2,10,11,27)(3,16,17,14)(6,22,26,29)(7,15,24,9)(8,23,25,28)(12,19,20,31)(13,18,30,32)Polycyclic presentation:
2^9|a^2=g,c^2=d,f^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,h^a=hi,c^b=cf,e^b=eh,f^b=fi,g^b=gh,e^c=eh,f^c=fi,g^c=gh,h^c=hi,e^d=ei,g^d=gi,f^e=fi,g^e=gi,g^f=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72341850 |
| maximal normal subgroup of | #72341850 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55664);