Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30437 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
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×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1065 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)(7,15)(8,20)(9,11)(10,25)(12,17)(13,22)(14,27)(16,30)(18,24)(19,29)(21,26)(23,31)(28,35)(32,34)(33,36);(5,7)(6,12)(8,16)(9,24)(10,18)(11,19)(13,30)(14,29)(15,25)(17,27)(20,36)(21,28)(22,34)(23,35)(26,33)(31,32);(5,8,11,23)(6,13,9,21)(7,16,19,35)(10,26,27,22)(12,28,24,30)(14,31,25,20)(15,32,29,36)(17,34,18,33);(5,9,11,6)(7,15)(8,21,23,13)(10,25,27,14)(12,18)(16,32)(17,24)(19,29)(20,22,31,26)(28,33)(30,34)(35,36)Polycyclic presentation:
2^9|a^2=h,c^2=i,d^2=ef,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,e^c=ei,f^c=fi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 292057);