Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23391 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1210 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 32 points:
(5,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,18,22,26)(19,27,29,24)(20,23,30,32)(21,31,28,25);(1,2)(3,4)(10,13)(14,16)(17,19,28,30,22,29,21,20)(18,23,31,24,26,32,25,27);(2,4)(9,11,12,15)(10,14,13,16)(17,20,21,29,22,30,28,19)(18,24,25,23,26,27,31,32);(5,7,8,6)(19,29)(20,30)(23,32)(24,27)Polycyclic presentation:
2^9|a^2=h,b^2=fi,c^2=e,d^2=hi,e^2=i,f^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,c^b=cg,d^b=di,d^c=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72264942, #72265013, #72265097, … |
| maximal normal subgroup of | #72264942, #72265013, #72265097, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266455);