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×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 | #60422 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1447 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)(7,13,8,20)(9,24,26,32)(10,15)(11,16)(12,19,29,23)(14,25,31,27)(17,21)(18,28)(22,36)(30,34)(33,35);(1,2,4,3)(5,7,10,23,11,8,21,19)(6,12,15,20,16,29,17,13)(9,18,27,33,26,22,25,34)(14,28,32,35,31,36,24,30);(5,8)(6,13)(7,11)(9,22)(10,19)(12,15)(14,30)(16,20)(17,29)(18,26)(21,23)(24,36)(25,33)(27,34)(28,32)(31,35);(1,3,4,2)(5,9,11,26)(6,14,16,31)(7,18,8,22)(10,27,21,25)(12,28,29,36)(13,30,20,35)(15,32,17,24)(19,34,23,33)Polycyclic presentation:
2^9|a^2=h,b^2=eh,d^2=efg,e^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,g^b=gi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 279928);