Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30496 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2669 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14357 |
| prime radical | C2×#3018 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1935 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)(3,15,18,27)(4,19)(5,23,25,11)(6,12)(7,13)(8,14)(9,16,28,17)(10,21)(20,32,24,29)(22,30)(26,31);(1,3)(2,9)(4,20)(5,21)(6,17)(7,16)(8,18)(10,29)(11,30)(12,27)(13,15)(14,28)(19,23)(22,24)(25,26)(31,32);(1,4)(2,10)(3,5)(6,21)(7,26)(8,22)(9,11)(12,30)(13,19)(14,31)(15,29)(16,20)(17,24)(18,25)(23,28)(27,32);(1,5)(2,11)(3,4)(6,24)(7,20)(8,25)(9,10)(12,32)(13,29)(14,23)(15,19)(16,26)(17,21)(18,22)(27,30)(28,31)Polycyclic presentation:
2^9|a^2=h,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,f^c=fi,g^c=gi,e^d=ei,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72215981, #72219685, #72219689, … |
| maximal normal subgroup of | #72215981, #72219685, #72219689, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290262);