Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #383160 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14357 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60367 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1981 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,12,7,10,6,11)(3,13,15,23,17,27,16,8)(4,18,20,9,21,26,19,25)(14,22,30,32,31,24,29,28);(1,3)(2,8)(4,14)(5,15)(6,16)(7,17)(9,24)(10,23)(11,27)(12,13)(18,32)(19,29)(20,30)(21,31)(22,25)(26,28);(1,4)(2,9)(3,14)(5,19)(6,20)(7,21)(8,22)(10,25)(11,26)(12,18)(13,28)(15,29)(16,30)(17,31)(23,24)(27,32);(2,10)(3,15)(5,6)(8,13)(9,25)(14,29)(16,17)(19,20)(22,28)(23,27)(24,32)(30,31)Polycyclic presentation:
2^9|a^2=h,e^2=i,f^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,e^b=ei,f^b=fi,f^c=fi,h^c=hi,e^d=ei,f^d=fi,h^d=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72216071, #72216154, #72216181, … |
| maximal normal subgroup of | #72216071, #72216154, #72216181, … |
| maximal quotient of | C2×#383162, #72252159, #72252192, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290358);