Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418458 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2424 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3068 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1575 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)(7,18)(8,14)(9,13)(10,25)(11,16)(12,17)(15,20)(19,34)(21,33)(22,27)(23,31)(24,32)(26,30)(28,35)(29,36);(5,7)(6,13)(8,22)(9,11)(10,20)(12,21)(14,28)(15,17)(16,33)(18,25)(19,23)(24,36)(26,35)(27,31)(29,30)(32,34);(2,4)(5,8,12,24)(6,14,17,32)(7,19,21,35)(9,22,20,36)(10,23,11,26)(13,27,15,29)(16,30,25,31)(18,34,33,28);(5,9,11,7,12,20,10,21)(6,15,25,18,17,13,16,33)(8,22,26,19,24,36,23,35)(14,29,31,34,32,27,30,28)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=ei,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,g^c=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 327076);