Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #219532 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2328 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3154 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1609 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,13,9,28)(8,22)(10,18)(11,17)(12,16)(14,26)(15,20,30,21)(19,32,24,31)(23,36,35,29)(25,33)(27,34);(5,7)(6,13)(8,23)(9,10)(11,21)(12,20)(14,31)(15,17)(16,30)(18,28)(19,27)(22,36)(24,26)(25,35)(29,33)(32,34);(2,4)(5,8,12,27,10,25,11,26)(6,14,17,33,18,34,16,22)(7,19,21,23,9,24,20,35)(13,29,30,31,28,36,15,32);(2,4)(5,9,10,7)(6,15)(8,24,25,19)(11,21,12,20)(13,17)(14,32)(16,28)(18,30)(22,36)(23,27,35,26)(29,33)(31,34)Polycyclic presentation:
2^9|a^2=h,c^2=g,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293277);