Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395174 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2487 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
886 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,10,17,12,18,11,16)(7,19,22,13,23,35,25,32)(8,24,26,14,21,36,30,33)(9,27,28,15,29,34,20,31);(1,2,3,4)(5,7,10,22,12,23,11,25)(6,13,16,19,18,32,17,35)(8,20,26,9,21,28,30,29)(14,15,24,27,33,31,36,34);(5,8)(6,14)(7,20)(9,22)(10,26)(11,30)(12,21)(13,31)(15,32)(16,24)(17,36)(18,33)(19,34)(23,28)(25,29)(27,35);(5,9,10,28,12,29,11,20)(6,15,16,27,18,31,17,34)(7,21,22,30,23,8,25,26)(13,24,19,33,32,36,35,14)Polycyclic presentation:
2^9|a^2=gh,b^2=e,d^2=h,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,f^b=fi,d^c=di,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72265193, #72265212, #72308040, … |
| maximal normal subgroup of | #72265193, #72265212, #72308040, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 322675);