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 | #2366 |
| prime radical | #23421 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1654 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)(3,4)(5,6,11,16,12,18,10,17)(7,19,22,13,23,25,32,34)(8,24,26,14,27,35,33,21)(9,28,29,15,31,20,30,36);(2,4)(5,7)(6,13)(8,15)(9,21)(10,32)(11,22)(12,23)(14,31)(16,25)(17,19)(18,34)(20,26)(24,29)(27,36)(28,33)(30,35);(5,8)(6,14)(7,20)(9,25)(10,33)(11,26)(12,27)(13,30)(15,32)(16,35)(17,24)(18,21)(19,31)(22,36)(23,28)(29,34);(2,4)(5,9,11,29,12,31,10,30)(6,15,16,20,18,36,17,28)(7,21,22,24,23,14,32,35)(8,25,26,34,27,19,33,13)Polycyclic presentation:
2^9|a^2=h,d^2=h,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,c^b=cg,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi,e^d=ei,f^d=fi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 282970);