Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31077 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | H(3,𝔽4) |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2684 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1170 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)(7,19)(8,24)(9,15)(10,17)(11,16)(12,18)(13,22)(14,26)(20,35)(21,32)(23,30)(25,34)(27,33)(28,31)(29,36);(5,7)(6,13)(8,25)(9,20)(10,28)(11,22)(12,23)(14,32)(15,34)(16,30)(17,19)(18,31)(21,27)(24,29)(26,36)(33,35);(5,8,10,9,12,27,11,26)(6,14,16,33,18,15,17,24)(7,20,22,25,23,36,28,21)(13,29,19,34,31,35,30,32);(5,9,10,27,12,26,11,8)(6,15,17,33,18,14,16,24)(7,21,22,20,23,25,28,36)(13,29,30,32,31,35,19,34)Polycyclic presentation:
2^9|a^2=h,c^2=e,d^2=g,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gi,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 294631);