Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #214688 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2308 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2220 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)(7,15,18,11)(8,19)(9,22)(10,13)(12,20)(14,24)(16,31,33,36)(17,26,35,30)(21,28)(23,27)(25,32,29,34);(1,2)(3,4)(5,7)(6,11)(8,16)(9,17)(10,18)(12,25)(13,15)(14,26)(19,36)(20,32)(21,33)(22,30)(23,34)(24,35)(27,29)(28,31);(2,4)(5,8,10,21)(6,12,13,27)(7,16,18,33)(9,23,24,20)(11,25,15,29)(14,28,22,19)(17,34,35,32)(26,31,30,36);(9,24)(12,27)(14,22)(17,35)(20,23)(25,29)(26,30)(32,34)Polycyclic presentation:
2^9|a^2=h,c^2=i,f^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,c^b=cg,h^b=hi,d^c=de,f^c=fi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 287019);