Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30437 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1421 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)(7,16)(8,21)(9,13)(10,15)(11,19)(12,24)(14,26)(17,30)(18,28)(20,29)(22,36)(23,31)(25,32)(27,34)(33,35);(5,7)(6,11)(8,22)(9,18)(10,20)(12,30)(13,16)(14,29)(15,28)(17,23)(19,26)(21,35)(24,34)(25,33)(27,32)(31,36);(5,8,10,24)(6,12)(7,17,20,33)(9,25,26,23)(11,27)(13,31)(14,32)(15,21)(16,30)(18,22,19,34)(28,36)(29,35);(6,13)(7,18)(9,26)(12,31)(14,15)(16,29)(17,22)(19,20)(21,32)(23,25)(30,35)(33,34)Polycyclic presentation:
2^9|a^2=h,c^2=eg,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi,e^d=ei,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293115);