Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24289 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | D4×C4⋊C4 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
992 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6,12,16,11,17,10,27)(7,14,20,29,8,21,19,13)(9,23,26,31,25,36,24,15)(18,32,34,35,22,28,33,30);(1,3,4,2)(5,7,12,20,11,8,10,19)(6,13,27,21,17,29,16,14)(9,18,26,34,25,22,24,33)(15,28,36,35,31,32,23,30);(2,3)(5,8)(6,14)(7,11)(9,22)(10,20)(12,19)(13,27)(15,30)(16,29)(17,21)(18,25)(23,32)(24,34)(26,33)(28,36)(31,35);(5,9,10,24,11,25,12,26)(6,15,16,23,17,31,27,36)(7,18,19,33,8,22,20,34)(13,28,14,30,29,32,21,35)Polycyclic presentation:
2^9|a^2=h,b^2=ef,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gi,g^c=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 267684);