Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #376201 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14175 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23439 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1532 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)(3,5)(4,17,21,29)(6,24)(7,12)(8,13)(9,15)(10,20,28,19)(11,22)(14,25,31,26)(16,23)(18,30,32,27);(1,3,6,15,8,16,7,11)(2,9,12,5,13,22,24,23)(4,14,19,27,21,31,20,30)(10,26,17,18,28,25,29,32);(1,4)(2,10)(3,14)(5,18)(6,20)(7,19)(8,21)(9,25)(11,27)(12,29)(13,28)(15,30)(16,31)(17,24)(22,26)(23,32);(1,5,6,22,8,23,7,9)(2,11,12,3,13,15,24,16)(4,18,20,26,21,32,19,25)(10,27,29,14,28,30,17,31)Polycyclic presentation:
2^9|a^2=h,b^2=e,d^2=eh,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,g^b=gi,e^c=ei,f^c=fi,h^c=hi,f^d=fi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72319525 |
| maximal normal subgroup of | #72319525 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 283397);