Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #364604 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13967 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3019 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2410 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,10,7,12,5,11)(3,13,16,8,17,24,22,25)(4,18,19,9,21,27,20,26)(14,30,32,29,15,23,31,28);(1,3,6,16,7,17,5,22)(2,8,10,24,12,25,11,13)(4,15,19,31,21,14,20,32)(9,23,27,28,26,30,18,29);(1,4)(2,9)(3,14)(5,19)(6,20)(7,21)(8,23)(10,18)(11,27)(12,26)(13,28)(15,17)(16,31)(22,32)(24,29)(25,30);(1,4)(2,9)(3,15)(5,20)(6,19)(7,21)(8,23)(10,27)(11,18)(12,26)(13,29)(14,17)(16,31)(22,32)(24,28)(25,30)Polycyclic presentation:
2^9|a^2=h,b^2=h,e^2=i,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,g^b=gi,f^c=fi,h^c=hi,e^d=ei77 conjugacy classes — view the full character table.
| maximal subgroup of | #72215931, #72215968, #72215974, … |
| maximal normal subgroup of | #72215931, #72215968, #72215974, … |
| maximal quotient of | C2×#364606 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271802);