Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23882 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2403 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13967 |
| prime radical | C2×#3025 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2428 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,12,8,13,6,24)(3,9,16,5,11,22,15,23)(4,17,21,10,20,29,25,28)(14,30,31,26,27,19,18,32);(1,3,7,16,8,11,6,15)(2,9,12,5,13,22,24,23)(4,18,21,14,20,31,25,27)(10,19,29,32,28,30,17,26);(1,4)(2,10)(3,14)(5,19)(6,25)(7,21)(8,20)(9,26)(11,27)(12,29)(13,28)(15,18)(16,31)(17,24)(22,32)(23,30);(1,5)(2,11)(3,13)(4,19)(6,9)(7,22)(8,23)(10,27)(12,15)(14,28)(16,24)(17,31)(18,29)(20,30)(21,32)(25,26)Polycyclic presentation:
2^9|a^2=h,b^2=h,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,g^b=gi,e^c=ei,f^c=fi,g^c=gi,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72237070, #72237072, #72237661, … |
| maximal normal subgroup of | #72237070, #72237072, #72237661, … |
| maximal quotient of | #72237711, C2×#364540, #72312200, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271736);