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 | 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 | |
896 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,11,7,10,5,20)(3,12,14,23,15,8,21,24)(4,16,18,25,19,31,17,9)(13,27,28,26,29,32,30,22);(1,3)(2,8)(4,13)(5,21)(6,14)(7,15)(9,22)(10,12)(11,23)(16,27)(17,28)(18,30)(19,29)(20,24)(25,26)(31,32);(1,4)(2,9,10,25)(3,13)(5,18,6,17)(7,19)(8,22,12,26)(11,16)(14,30,21,28)(15,29)(20,31)(23,32)(24,27)Polycyclic presentation:
2^9|a^2=g,c^2=d,g^2=i,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 | #72238713, #72238717, #72238722, … |
| maximal normal subgroup of | #72238713, #72238717, #72238722, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55663);