Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#540 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2960 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,4)(5,6)(7,8)(17,18,22,24,21,26,20,25)(19,23,29,32,27,31,28,30);(2,4)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,28)(21,27)(22,29)(24,32)(25,30)(26,31);(6,8)(10,13)(14,16)(18,24)(19,27)(20,22)(23,30)(25,26)(31,32);(2,4)(9,11,12,15)(10,14,13,16)(19,27)(23,31)(28,29)(30,32)Polycyclic presentation:
2^9|a^2=h,d^2=gi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=di,f^c=fi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72265008, #72265020, #72265029, … |
| maximal normal subgroup of | #72265008, #72265020, #72265029, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 270362);