Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418518 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2416 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1679 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)(7,19)(8,23)(9,15)(10,17)(11,16)(12,18)(13,21)(14,27)(20,29)(22,30)(24,36)(25,31)(26,34)(28,33)(32,35);(5,7)(6,13)(8,24)(9,10)(11,21)(12,22)(14,31)(15,18)(16,30)(17,19)(20,26)(23,32)(25,28)(27,35)(29,33)(34,36);(2,4)(5,8)(6,14)(7,20)(9,25)(10,26)(11,27)(12,28)(13,29)(15,32)(16,33)(17,23)(18,34)(19,31)(21,24)(22,35)(30,36);(5,9,11,22,12,21,10,7)(6,15,17,19,18,13,16,30)(8,25,27,35,28,24,26,20)(14,32,23,31,34,29,33,36)Polycyclic presentation:
2^9|d^2=ei,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,f^c=fi,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, 325715);