Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #442706 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15727 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
855 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,8,14,7,3,6,11)(4,12,16,27,17,9,21,24)(5,10,22,18,20,13,19,25)(15,26,29,31,30,23,32,28);(1,3,7,2)(4,9)(5,18,20,25)(6,11,8,14)(10,19,13,22)(12,17)(15,28)(16,27)(21,24)(23,29)(26,32)(30,31);(1,4,6,16,7,17,8,21)(2,9,11,24,3,12,14,27)(5,15,19,29,20,30,22,32)(10,23,25,28,13,26,18,31);(1,5)(2,10,3,13)(4,15)(6,19)(7,20)(8,22)(9,23,12,26)(11,25,14,18)(16,29)(17,30)(21,32)(24,28,27,31)Polycyclic presentation:
2^9|a^2=efg,b^2=fg,c^2=e,d^2=f,e^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,g^c=gi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72339677, #72339680, #72339684 |
| maximal normal subgroup of | #72339677, #72339680, #72339684 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 349902);