Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417570 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2580 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
977 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,18)(8,9,24,14)(10,28)(11,16)(12,17)(13,21)(15,27,29,26)(19,35,36,20)(22,31)(23,34)(25,33,30,32);(1,2)(3,4)(5,7,10,21,12,23,11,22)(6,13,16,18,17,31,28,34)(8,19,15,32,24,36,29,33)(9,25,26,35,14,30,27,20);(2,4)(5,8,10,15,12,24,11,29)(6,14,28,26,17,9,16,27)(7,19,21,32,23,36,22,33)(13,30,34,35,31,25,18,20);(5,9)(6,15)(7,20)(8,16)(10,27)(11,26)(12,14)(13,19)(17,29)(18,33)(21,30)(22,25)(23,35)(24,28)(31,36)(32,34)Polycyclic presentation:
2^9|a^2=efh,b^2=e,c^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,h^c=hi,e^d=ei,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72322946 |
| maximal normal subgroup of | #72322946 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 357484);