Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #383753 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14363 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60426 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1374 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11,7,8,5,20)(3,12,13,22,16,25,15,9)(4,17,18,23,19,26,21,10)(14,24,29,31,30,32,28,27);(2,8)(3,13)(5,6)(9,12)(14,28)(15,16)(17,26)(18,21)(22,25)(24,31)(27,32)(29,30);(1,3)(2,9,8,22)(4,14)(5,15)(6,13)(7,16)(10,24,23,32)(11,12,20,25)(17,31,26,27)(18,29)(19,30)(21,28);(1,4)(2,10,8,23)(3,14)(5,18)(6,21)(7,19)(9,24,22,32)(11,26,20,17)(12,27,25,31)(13,28)(15,29)(16,30)Polycyclic presentation:
2^9|a^2=h,c^2=e,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^d=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72213409, #72216166, #72216192, … |
| maximal normal subgroup of | #72213409, #72216166, #72216192, … |
| maximal quotient of | #72237183, #72250493, C2×#383755 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290951);