Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476705 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17095 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2008 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1038 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,12,7,13,23,11)(3,14,17,8,18,27,32,28)(4,19,5,10,21,29,24,9)(15,26,16,31,20,25,22,30);(1,3,6,17,7,18,23,32)(2,8,11,14,13,28,12,27)(4,20,5,22,21,15,24,16)(9,25,29,31,10,26,19,30);(1,4)(2,9)(3,15)(5,23)(6,24)(7,21)(8,25)(10,13)(11,19)(12,29)(14,30)(16,32)(17,22)(18,20)(26,28)(27,31);(1,5,23,4,7,24,6,21)(2,10,12,29,13,9,11,19)(3,16,32,15,18,22,17,20)(8,26,27,31,28,25,14,30)Polycyclic presentation:
2^9|a^2=f,b^2=e,d^2=ei,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=def,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,h^c=hi,h^d=hi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72251688, #72251691, #72251697, … |
| maximal normal subgroup of | #72251688, #72251691, #72251697, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 383902);