Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #148465 |
| 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 | C22×#41 | C23 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
820 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)(3,12,15,29)(4,10,17,7)(5,18,19,9)(6,11)(8,14)(13,27)(16,23)(20,24)(21,26)(22,30,25,32)(28,31);(2,7)(3,13)(5,19)(8,24)(9,23)(10,11)(14,30)(15,31)(16,21)(18,26)(20,32)(22,25);(1,3,4,13)(2,8,7,22)(5,20,21,30)(6,15,17,31)(9,12,26,27)(10,25,11,24)(14,16,32,19)(18,29,23,28)Polycyclic presentation:
2^9|a^2=g,c^2=d,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fhi,h^a=hi,c^b=cf,e^b=ehi,g^b=gh,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 | #72303347, #72303352, #72303358 |
| maximal normal subgroup of | #72303347, #72303352, #72303358 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55661);