Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #434065 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15496 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23402 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1212 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,20,5,10,6,11)(3,12,21,24,14,8,15,28)(4,16,18,31,19,9,22,26)(13,27,17,30,29,23,32,25);(1,3,5,14)(2,8,10,12)(4,17)(6,15,7,21)(9,25)(11,24,20,28)(13,22)(16,30)(18,29)(19,32)(23,31)(26,27);(1,4)(2,9,10,16)(3,13,14,29)(5,19)(6,18)(7,22)(8,23)(11,26,20,31)(12,27)(15,17,21,32)(24,25)(28,30);(2,10)(3,14)(6,7)(9,16)(13,29)(18,22)(24,28)(25,30)Polycyclic presentation:
2^9|a^2=eg,b^2=e,c^2=ef,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,f^b=fi,e^c=ei,f^c=fi,g^d=gi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72287274, #72287280, #72287354, … |
| maximal normal subgroup of | #72287274, #72287280, #72287354, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 341261);