Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #459491 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16341 |
| 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 | |
1050 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11,7,12,5,9)(3,13,16,24,17,23,14,8)(4,18,20,10,21,25,19,26)(15,22,30,27,29,32,28,31);(1,3)(2,8,12,24)(4,15)(5,14)(6,16)(7,17)(9,23,11,13)(10,22,26,32)(18,31,25,27)(19,28)(20,30)(21,29);(2,9)(3,14)(4,19)(5,6)(11,12)(13,23)(15,29)(16,17)(18,25)(20,21)(22,27)(31,32);(1,4,6,20,7,21,5,19)(2,10,11,25,12,26,9,18)(3,15,16,30,17,29,14,28)(8,22,13,27,24,32,23,31)Polycyclic presentation:
2^9|a^2=f,b^2=ef,d^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,g^b=gi,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72339020, #72339031, #72339032, … |
| maximal normal subgroup of | #72339020, #72339031, #72339032, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 366687);