Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #460311 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2500 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #16816 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
899 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)(3,6)(4,9)(5,19)(7,12)(8,11)(10,22)(13,24)(14,23)(15,17)(16,27)(18,26)(20,25)(21,28)(29,30)(31,32);(1,3,6,12,7,11,8,2)(4,15,17,26,18,13,24,9)(5,14,22,28,23,19,21,10)(16,27,31,32,30,29,20,25);(1,4,8,17,7,18,6,24)(2,9,3,13,12,26,11,15)(5,20,21,16,23,31,22,30)(10,27,14,25,28,29,19,32);(1,5)(2,10)(3,14)(4,16)(6,22)(7,23)(8,21)(9,25)(11,19)(12,28)(13,29)(15,27)(17,31)(18,30)(20,24)(26,32)Polycyclic presentation:
2^9|b^2=gh,c^2=ef,e^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,e^b=ei,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,e^d=ei,f^d=fi,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 379476);