Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23886 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
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×C22⋊C4 |
| prime radical | #60422 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1696 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)(5,6,11,16,12,18,10,17)(7,19,22,13,24,33,23,34)(8,25,28,14,21,32,27,35)(9,29,30,15,20,31,26,36);(5,7,11,22,12,24,10,23)(6,13,16,33,18,34,17,19)(8,26,28,9,21,30,27,20)(14,15,32,31,35,36,25,29);(1,2,3,4)(5,8)(6,14)(7,20)(9,24)(10,27)(11,28)(12,21)(13,31)(15,19)(16,32)(17,25)(18,35)(22,26)(23,30)(29,34)(33,36);(5,9)(6,15)(7,21)(8,24)(10,26)(11,30)(12,20)(13,32)(14,19)(16,31)(17,29)(18,36)(22,27)(23,28)(25,34)(33,35)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=fg,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 276101);