Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25196 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2424 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14112 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
982 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)(3,7,13,21)(4,8,15,24)(5,17)(6,9)(10,18)(11,22,14,19)(12,26)(16,30,29,23)(20,27)(25,31)(28,32);(1,3)(2,7,9,21)(4,12)(5,11)(6,13)(8,20,24,26)(10,19,17,22)(14,18)(15,27)(16,25)(23,31,30,32)(28,29);(3,11,13,14)(7,19,21,22)(12,25,27,28)(20,31,26,32);(1,4,5,16,6,15,18,29)(2,8,10,23,9,24,17,30)(3,12,11,25,13,27,14,28)(7,20,19,31,21,26,22,32)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=f,d^2=fg,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,g^b=gi,h^b=hi,e^d=ei,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72213889, #72219485, #72219486 |
| maximal normal subgroup of | #72213889, #72219485, #72219486 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 280410);