Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #222240 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1441 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6,10,14,11,16,9,15)(7,17,21,12,20,28,19,29)(8,22,24,30,26,31,25,13)(18,33,34,32,23,27,35,36);(1,2,4,3)(5,7)(6,12)(8,23,26,18)(9,21)(10,19)(11,20)(13,27,30,33)(14,17)(15,28)(16,29)(22,32,31,36)(24,34,25,35);(5,8)(6,13)(7,18)(9,25)(10,24)(11,26)(12,27)(14,22)(15,31)(16,30)(17,32)(19,35)(20,23)(21,34)(28,36)(29,33);(1,3,4,2)(6,14)(9,10)(12,17)(13,22)(15,16)(19,21)(24,25)(27,32)(28,29)(30,31)(33,36)(34,35)Polycyclic presentation:
2^9|a^2=h,b^2=eh,d^2=efg,e^2=i,f^2=i,h^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,f^c=fi,e^d=ei,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 279937);