Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419895 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1140 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6)(7,17,8,19)(9,13)(10,16)(11,15)(12,18,14,20);(5,7,9,18,10,8,11,20)(6,12,15,19,16,14,13,17);(6,13)(9,11)(12,19)(14,17)(15,16)(18,20);(2,4)(5,8)(6,14,16,12)(7,10)(9,18)(11,20)(13,17,15,19)Polycyclic presentation:
2^9|a^2=efg,b^2=f,d^2=efi,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,g^b=gi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72338548, #72338552, #72338559 |
| maximal normal subgroup of | #72338548, #72338552, #72338559 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 344164);