Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31077 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C2×C42 |
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 | #2804 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
822 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)(7,17)(8,22)(9,12)(10,15)(11,16)(13,20)(14,24)(18,27)(19,31)(21,29)(23,26)(25,30)(28,33)(32,35)(34,36);(6,12,16,15)(7,18,21,20)(8,23,24,25)(13,27)(14,22)(17,29)(26,30)(28,36,35,31);(5,7,9,18,11,21,10,20)(6,13,12,17,16,27,15,29)(8,19,25,32,24,34,23,33)(14,28,26,31,22,35,30,36);(5,8)(6,14)(7,19)(9,25)(10,23)(11,24)(12,26)(13,28)(15,30)(16,22)(17,31)(18,32)(20,33)(21,34)(27,35)(29,36)Polycyclic presentation:
2^9|a^2=h,b^2=eg,c^2=efg,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,f^c=fi,g^c=gi,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 294428);