Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #222240 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×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 | C2×#3010 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2215 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,11,16,12,17,10,29)(7,18,21,13,22,33,30,34)(8,23,25,20,26,14,32,35)(9,19,27,15,28,24,31,36);(1,2,3,4)(5,7,10,30,12,22,11,21)(6,13,29,18,17,34,16,33)(8,24,32,15,26,19,25,36)(9,20,31,23,28,35,27,14);(2,4)(5,8)(6,14)(7,19)(9,13)(10,25)(11,32)(12,26)(15,30)(16,20)(17,23)(18,27)(21,36)(22,24)(28,34)(29,35)(31,33);(5,9,12,28)(6,15,17,36)(7,20,22,35)(8,13,26,34)(10,31,11,27)(14,30,23,21)(16,24,29,19)(18,25,33,32)Polycyclic presentation:
2^9|a^2=h,b^2=fghi,d^2=i,e^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,h^c=hi,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 273924);