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 |
| prime radical | C2×#2008 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1447 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)(6,12)(8,20)(9,10)(13,26)(15,25)(16,29)(17,21)(18,24)(22,23)(27,31)(30,33)(34,35);(5,6,9,12,11,15,10,25)(7,16,18,29,19,26,24,13)(8,14,20,27,23,32,22,31)(17,33,35,36,34,30,21,28);(2,4)(5,7,11,19)(6,13,15,29)(8,21,23,35)(9,24,10,18)(12,26,25,16)(14,30,32,33)(17,20,34,22)(27,36,31,28);(5,8)(6,14)(7,17)(9,20)(10,22)(11,23)(12,27)(13,28)(15,32)(16,33)(18,35)(19,34)(21,24)(25,31)(26,30)(29,36)Polycyclic presentation:
2^9|b^2=g,c^2=i,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, 371277);