Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #364600 |
| 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 | #60369 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2389 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)(3,13)(4,17,22,28)(5,15)(6,11)(7,10)(8,12)(9,21,27,20)(14,30,32,19)(16,23)(18,26,31,29)(24,25);(1,3)(2,5)(4,18,22,31)(6,25)(7,15)(8,16)(9,26,27,29)(10,23)(11,13)(12,24)(14,20,32,21)(17,30,28,19);(1,4)(2,9)(3,14)(5,19)(6,20)(7,21)(8,22)(10,17)(11,28)(12,27)(13,29)(15,31)(16,32)(18,25)(23,26)(24,30);(1,5)(2,3)(4,19)(6,23)(7,13)(8,24)(9,14)(10,25)(11,15)(12,16)(17,18)(20,26)(21,29)(22,30)(27,32)(28,31)Polycyclic presentation:
2^9|a^2=h,b^2=h,e^2=i,f^2=i,g^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,f^c=fi,g^c=gi,h^c=hi,e^d=ei77 conjugacy classes — view the full character table.
| maximal subgroup of | #72215936, #72216071, #72216123, … |
| maximal normal subgroup of | #72215936, #72216071, #72216123, … |
| maximal quotient of | C2×#364600, #72239623, #72239723, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271796);