Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #363956 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13956 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #62855 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2573 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,9,7,11,5,12)(3,8,16,20,14,23,15,22)(4,10,18,25,17,26,19,24)(13,27,28,31,29,21,30,32);(1,3)(2,8,11,23)(4,13)(5,16)(6,15)(7,14)(9,22,12,20)(10,21,26,27)(17,29)(18,30)(19,28)(24,32,25,31);(2,9)(5,6)(8,20)(10,24)(11,12)(15,16)(18,19)(21,31)(22,23)(25,26)(27,32)(28,30);(1,4)(2,10,11,26)(3,13)(5,18)(6,19)(7,17)(8,21,23,27)(9,24,12,25)(14,29)(15,28)(16,30)(20,31,22,32)Polycyclic presentation:
2^9|a^2=h,b^2=g,d^2=g,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,f^c=fi,h^c=hi,e^d=ei,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72233772, #72233773, #72233779, … |
| maximal normal subgroup of | #72233772, #72233773, #72233779, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271154);