Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30428 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×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×C22⋊C4 |
| prime radical | #60643 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1605 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)(8,22,26,27)(9,15,29,36)(10,30)(11,16)(12,17)(13,20)(14,25,35,32)(19,28,23,33)(21,24)(31,34);(2,4)(5,7)(6,13)(8,23)(9,27)(10,31)(11,20)(12,21)(14,28)(15,32)(16,24)(17,34)(18,30)(19,26)(22,29)(25,36)(33,35);(1,2,3,4)(5,8,12,26)(6,14,17,35)(7,19,21,23)(9,24,29,18)(10,25,11,32)(13,33,34,28)(15,20,36,31)(16,27,30,22);(5,9,10,28,12,29,11,33)(6,15,16,19,17,36,30,23)(7,14,20,27,21,35,31,22)(8,24,25,13,26,18,32,34)Polycyclic presentation:
2^9|a^2=h,c^2=g,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,e^b=ei,f^c=fi,h^c=hi,f^d=fi,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293268);