Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30437 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1505 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4)(5,6)(7,18)(8,22)(9,14)(10,16)(11,15)(12,17)(13,24)(19,29)(20,27)(21,26)(23,34)(25,32)(28,31)(30,33)(35,36);(5,7)(6,9)(8,19,25,34)(10,20)(11,14)(12,21)(13,29,31,23)(15,26)(16,18)(17,27)(22,33,32,36)(24,35,28,30);(5,8)(6,13)(7,19)(9,23)(10,24)(11,28)(12,25)(14,30)(15,22)(16,32)(17,31)(18,33)(20,35)(21,34)(26,36)(27,29);(5,9,12,27)(6,14,17,20)(7,16,21,15)(8,23,25,29)(10,26,11,18)(13,30,31,35)(19,32,34,22)(24,36,28,33)Polycyclic presentation:
2^9|a^2=h,b^2=ef,d^2=i,e^2=i,f^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,e^b=ei,f^b=fi,g^b=gi,f^c=fi,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, 291122);