Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24292 |
| 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 | #2361 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1202 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,12,16)(7,11,15,10)(8,17)(9,18,25,23)(13,27)(14,26,30,24)(19,21)(20,32)(22,29)(28,35)(31,33)(34,36);(1,2)(3,4)(5,7,11,6,12,15,10,16)(8,19,21,29,22,13,27,17)(9,23,24,30,25,18,26,14)(20,28,34,31,35,32,33,36);(2,4)(5,8,10,21,12,22,11,27)(6,13,7,17,16,19,15,29)(9,20,26,34,25,35,24,33)(14,28,18,31,30,32,23,36);(5,9)(6,14)(7,18)(8,20)(10,26)(11,24)(12,25)(13,28)(15,23)(16,30)(17,31)(19,32)(21,34)(22,35)(27,33)(29,36)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=fh,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, 281893);