Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #386085 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14419 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60644 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1594 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,8,13,6,12,7,22)(3,14,4,10,17,9,16,25)(5,19,20,28,21,11,23,29)(15,27,31,32,30,24,18,26);(1,3)(2,9)(4,7)(5,15)(6,17)(8,16)(10,13)(11,27)(12,14)(18,20)(19,24)(21,30)(22,25)(23,31)(26,29)(28,32);(1,4,8,17,6,16,7,3)(2,10,22,14,12,25,13,9)(5,18,23,30,21,31,20,15)(11,26,29,27,19,32,28,24);(1,5)(2,11,12,19)(3,15)(4,18)(6,21)(7,20)(8,23)(9,24,14,27)(10,26,25,32)(13,28,22,29)(16,31)(17,30)Polycyclic presentation:
2^9|a^2=h,c^2=gi,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,f^c=fi,f^d=fi,h^d=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72339029, #72339037, #72339042 |
| maximal normal subgroup of | #72339029, #72339037, #72339042 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293284);