Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #493500 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #20585 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| prime radical | #62876 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #3498 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
2277 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,12)(4,16,18,9)(5,10)(6,11)(7,13,22,27)(8,17,24,19)(14,26)(15,21)(20,25)(23,28,32,30)(29,31);(1,3,6,14)(2,7,11,22)(4,13,18,27)(5,15,20,29)(8,21,24,31)(9,12,16,26)(10,23,25,32)(17,28,19,30);(1,4)(2,8)(3,13)(5,19)(6,18)(7,21)(9,10)(11,24)(12,23)(14,27)(15,30)(16,25)(17,20)(22,31)(26,32)(28,29);(3,14)(5,20)(7,22)(8,24)(10,25)(12,26)(13,27)(17,19)Polycyclic presentation:
2^9|a^2=g,b^2=i,f^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=di,g^b=gh,d^c=dh,f^c=fi,g^c=gi,f^d=fi,g^e=gi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72232878, #72260499, #72260545 |
| maximal normal subgroup of | #72232878, #72260499, #72260545 |
| maximal quotient of | #72232835, #72233271, #72238365, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400700);