Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455572 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1487 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)(7,13)(8,22)(9,16)(10,15)(11,17)(12,19)(14,24)(18,33)(20,29)(21,28)(23,36)(25,32)(26,31)(27,34)(30,35);(5,7,10,20,11,21,9,19)(6,12,16,28,17,29,15,13)(8,18,24,23,25,34,26,35)(14,27,22,36,31,33,32,30);(2,4)(5,7)(6,13)(8,23)(9,20)(10,19)(11,21)(12,15)(14,27)(16,29)(17,28)(18,24)(22,30)(25,35)(26,34)(31,33)(32,36);(5,8)(6,14)(7,18)(9,26)(10,24)(11,25)(12,27)(13,30)(15,32)(16,22)(17,31)(19,35)(20,23)(21,34)(28,36)(29,33)Polycyclic presentation:
2^9|b^2=g,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,h^b=hi,d^c=de,e^c=ei,g^c=gi,e^d=ei,f^d=fi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 371276);