Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24292 |
| 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 | #60366 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2230 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)(7,16)(8,21,25,32)(9,14)(10,15)(11,18)(12,24,30,23)(13,26)(17,27,34,36)(19,28)(20,31)(22,29,35,33);(1,2,3,4)(5,7,10,20)(6,11,15,28)(8,22)(9,19,26,18)(12,29)(13,31,14,16)(17,23)(21,36)(24,34)(25,35)(27,32)(30,33);(2,4)(5,8)(6,12)(7,17)(9,23)(10,25)(11,27)(13,21)(14,32)(15,30)(16,33)(18,22)(19,35)(20,34)(24,26)(28,36)(29,31);(5,7)(6,11)(8,17)(9,19)(10,20)(12,27)(13,31)(14,16)(15,28)(18,26)(21,29)(22,24)(23,35)(25,34)(30,36)(32,33)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,d^b=di,e^b=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 273910);