Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #22455 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | D4×C22⋊C4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13813 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1515 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,12,8,13,6,25)(3,14,19,31,17,9,18,28)(4,20,21,10,22,5,11,24)(15,23,27,30,32,16,29,26);(1,3)(2,9,13,14)(4,15)(5,23,20,16)(6,18)(7,19)(8,17)(10,26,24,30)(11,29)(12,28,25,31)(21,27)(22,32);(1,4,7,21,8,22,6,11)(2,10,12,5,13,24,25,20)(3,15,19,27,17,32,18,29)(9,26,28,23,14,30,31,16);(1,5,7,24,8,20,6,10)(2,11,12,4,13,21,25,22)(3,16,19,26,17,23,18,30)(9,27,28,32,14,29,31,15)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=g,d^2=g,f^2=i,g^2=i,b^a=be,c^a=cf,f^a=fi,h^a=hi,d^b=de,e^b=ei,f^c=fi,h^d=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72319403, #72319742, #72342961, … |
| maximal normal subgroup of | #72319403, #72319742, #72342961, … |
| maximal quotient of | #72342987 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 263841);