Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #185475 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2494 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,10,6,9,13,16)(3,8,11,15,4,7,12,14)(17,18)(19,26,28,23)(20,22)(21,24)(25,29)(27,31,32,30);(1,3,6,4)(2,7,9,8)(5,12,13,11)(10,15,16,14)(18,22)(20,29)(23,30)(24,25)(26,31)(27,32);(1,4,6,3)(2,8,9,7)(5,11,13,12)(10,14,16,15)(17,19)(18,23)(20,27)(21,28)(22,30)(24,26)(25,31)(29,32);(18,24)(22,25)(23,26)(30,31)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=hi,e^2=i,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,e^b=ei,g^b=ghi,g^c=gh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72257895, #72257904, #72281573, … |
| maximal normal subgroup of | #72257895, #72257904, #72281573, … |
| maximal quotient of | #72264381 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 92969);