Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #428863 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15391 |
| 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 | |
974 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)(3,13)(4,19,22,27)(5,11)(6,10)(7,12)(8,16)(9,21,28,23)(14,18)(15,30,31,24)(17,25)(20,26,32,29);(1,3,5,16,7,18,6,17)(2,8,10,13,12,25,11,14)(4,20,21,31,22,32,23,15)(9,26,27,30,28,29,19,24);(1,2)(3,14)(4,9)(5,10)(6,11)(7,12)(8,16)(13,18)(15,26)(17,25)(19,23)(20,30)(21,27)(22,28)(24,32)(29,31);(1,4,5,23,7,22,6,21)(2,9,10,19,12,28,11,27)(3,15,16,32,18,31,17,20)(8,24,13,29,25,30,14,26)Polycyclic presentation:
2^9|a^2=f,b^2=e,d^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,g^b=gi,g^c=gi,h^c=hi,e^d=ei,f^d=fi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72251783, #72251789, #72251800, … |
| maximal normal subgroup of | #72251783, #72251789, #72251800, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 336059);