Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #366639 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13995 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3021 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2219 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)(3,13,17,8)(4,18)(5,21,23,10)(6,24)(7,11)(9,20)(12,27)(14,22,31,26)(15,19)(16,28,29,32)(25,30);(1,3,6,16,7,17,27,29)(2,8,24,32,11,13,12,28)(4,14,20,10,19,31,25,21)(5,15,26,30,23,18,22,9);(1,4)(2,9)(3,14)(5,13)(6,25)(7,19)(8,23)(10,29)(11,30)(12,15)(16,21)(17,31)(18,24)(20,27)(22,28)(26,32);(1,5)(2,10)(3,15)(4,13)(6,26)(7,23)(8,19)(9,29)(11,21)(12,14)(16,30)(17,18)(20,28)(22,27)(24,31)(25,32)Polycyclic presentation:
2^9|a^2=h,b^2=fghi,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,g^b=gi,h^b=hi,e^c=ei,f^c=fi,f^d=fi,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72241168, #72241179, #72241304, … |
| maximal normal subgroup of | #72241168, #72241179, #72241304, … |
| maximal quotient of | C2×#366640 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 273836);