Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #478457 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2760 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C2×#534 | C22 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1140 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)(7,15,18,11)(8,12,21,28)(9,22,23,13)(10,14)(16,26)(17,25)(19,24)(20,27)(29,34)(30,32)(31,36,33,35);(5,7)(6,11)(8,19,21,16)(9,17)(10,18)(12,26,28,24)(13,25)(14,15)(20,31,34,33)(22,30)(23,32)(27,35,29,36);(5,8)(6,12,14,28)(7,16)(9,20)(10,21)(11,24,15,26)(13,29,22,27)(17,31)(18,19)(23,34)(25,36,30,35)(32,33);(2,4)(5,9,10,23)(6,13,14,22)(7,17,18,32)(8,20,21,34)(11,25,15,30)(12,27,28,29)(16,31,19,33)(24,35,26,36)Polycyclic presentation:
2^9|a^2=efg,b^2=ef,c^2=e,d^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=def,g^b=gi,d^c=de,f^c=fi,h^c=hi,g^d=gi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72285240, #72285288, #72285413, … |
| maximal normal subgroup of | #72285240, #72285288, #72285413, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 385654);