Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23886 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
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 | |
982 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6)(7,19,24,27)(8,25)(9,31,34,20)(10,16)(11,17)(12,18)(13,22,35,23)(14,28)(15,32,26,33)(21,29)(30,36);(1,2,4,3)(5,7,10,23,12,24,11,22)(6,13,16,27,18,35,17,19)(8,26,28,31,30,15,29,20)(9,21,33,25,34,14,32,36);(1,2,4,3)(5,8,10,28,12,30,11,29)(6,14,17,25,18,21,16,36)(7,20,22,15,24,31,23,26)(9,27,33,35,34,19,32,13);(1,3,4,2)(5,9)(6,15)(7,21)(8,27)(10,33)(11,32)(12,34)(13,29)(14,24)(16,20)(17,31)(18,26)(19,30)(22,36)(23,25)(28,35)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=f,d^2=fg,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,e^d=ei,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 280379);