Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23886 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
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×C4⋊C4 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1246 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)(5,6,10,16,11,17,18,15)(7,8,21,23,13,24,20,12)(9,25,27,32,28,31,26,14)(19,29,33,22,30,35,34,36);(5,7,18,20,11,13,10,21)(6,12,15,24,17,23,16,8)(9,19,26,34,28,30,27,33)(14,29,31,36,32,35,25,22);(2,4)(5,8,10,23,11,24,18,12)(6,13,15,21,17,7,16,20)(9,22,27,35,28,36,26,29)(14,30,31,33,32,19,25,34);(5,9)(6,14)(7,19)(8,22)(10,27)(11,28)(12,29)(13,30)(15,31)(16,25)(17,32)(18,26)(20,34)(21,33)(23,35)(24,36)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,c^2=g,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,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 277350);