Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419852 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1672 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,15,9,16,11,13)(7,12,17,27,19,28,20,31)(8,21,24,14,23,29,25,30)(18,33,34,26,22,35,32,36);(1,2)(3,4)(5,7)(6,12)(8,22)(9,19)(10,20)(11,17)(13,27)(14,26)(15,31)(16,28)(18,23)(21,35)(24,34)(25,32)(29,33)(30,36);(2,4)(6,13)(7,17)(10,11)(14,29)(15,16)(18,32)(19,20)(21,30)(22,34)(24,25)(27,31)(33,35);(5,8,9,23)(6,14)(7,18,19,22)(10,25,11,24)(12,26)(13,29)(15,21)(16,30)(17,32,20,34)(27,33)(28,36)(31,35)Polycyclic presentation:
2^9|a^2=f,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,f^c=fi,g^c=gi,h^c=hi,f^d=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72255658, #72256284, #72256291 |
| maximal normal subgroup of | #72255658, #72256284, #72256291 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 327048);