Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476297 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2002 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1205 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)(5,6,9,11,10,14,8,13)(7,15,18,20,16,19,17,12);(6,11)(7,16)(8,9)(12,19)(13,14)(15,20);(5,7,8,18,10,16,9,17)(6,12,13,15,14,20,11,19);(2,4)(6,11)(7,17)(8,9)(12,20)(13,14)(16,18)Polycyclic presentation:
2^9|a^2=f,c^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=def,e^b=ei,f^b=fi,h^b=hi,d^c=de,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72285442, #72285552, #72285571, … |
| maximal normal subgroup of | #72285442, #72285552, #72285571, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 383676);