Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454002 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2010 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2196 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)(8,22)(9,26)(10,16)(11,15)(12,17)(13,20)(14,24)(19,29)(21,30)(23,31)(25,32)(27,35)(28,33)(34,36);(5,7,12,9)(6,13,17,30)(8,23,25,34)(10,21,11,20)(14,31,33,36)(15,26,16,18)(19,28,27,24)(22,35,32,29);(2,4)(5,8)(6,14)(7,19)(9,27)(10,28)(11,24)(12,25)(13,29)(15,32)(16,22)(17,33)(18,31)(20,34)(21,23)(26,36)(30,35);(5,9,12,7)(6,13,17,30)(8,23,25,34)(10,20,11,21)(14,31,33,36)(15,26,16,18)(19,28,27,24)(22,35,32,29)Polycyclic presentation:
2^9|b^2=i,d^2=i,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,e^d=ei,f^d=fi,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72250742, #72250795, #72255916 |
| maximal normal subgroup of | #72250742, #72250795, #72255916 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361200);