Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #28182 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2535 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14068 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1133 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,10,7,11,5,22)(3,12,15,25,16,24,14,8)(4,17,19,9,21,27,20,28)(13,30,31,26,32,23,18,29);(1,3)(2,8,11,25)(4,18,21,31)(5,14)(6,15)(7,16)(9,26)(10,12,22,24)(13,20,32,19)(17,30)(23,27)(28,29);(1,4)(2,9)(3,13)(5,20)(6,19)(7,21)(8,23)(10,27)(11,28)(12,29)(14,18)(15,31)(16,32)(17,22)(24,26)(25,30);(1,3)(2,8,11,25)(4,13)(5,14)(6,15)(7,16)(9,23,28,30)(10,12,22,24)(17,26,27,29)(18,20)(19,31)(21,32)Polycyclic presentation:
2^9|a^2=h,b^2=eh,d^2=eg,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,f^b=fi,e^c=ei,f^c=fi,g^c=gi,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72287080, #72287274, #72287308, … |
| maximal normal subgroup of | #72287080, #72287274, #72287308, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 278386);