Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23472 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#124 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#400 |
| prime radical | C25 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1578 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)(5,6,11,15,10,14,9,24)(7,16,18,12,19,28,25,27)(8,13,21,32,22,30,23,31)(17,29,20,26,35,33,34,36);(2,4)(5,7,11,18,10,19,9,25)(6,12,15,28,14,27,24,16)(8,20,23,17,22,34,21,35)(13,29,31,36,30,33,32,26);(2,4)(5,8,11,23,10,22,9,21)(6,13,15,31,14,30,24,32)(7,17,18,34,19,35,25,20)(12,26,28,29,27,36,16,33);(6,14)(12,27)(13,30)(15,24)(16,28)(26,36)(29,33)(31,32)Polycyclic presentation:
2^9|a^2=h,b^2=ef,c^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265767);