Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #196332 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | #35088 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
2647 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)(7,15,18,11)(8,12,20,26)(9,21,22,13)(10,14)(16,23)(17,24)(19,25)(27,30)(28,31)(29,34,36,35)(32,33);(1,2)(3,4)(5,7)(6,11)(8,16)(9,17)(10,18)(12,23)(13,24)(14,15)(19,29)(20,30)(21,28)(22,31)(25,34)(26,27)(32,36)(33,35);(2,4)(5,8)(6,12,14,26)(7,16)(9,19)(10,20)(11,23,15,27)(13,25,21,33)(17,29)(18,30)(22,32)(24,34,28,35)(31,36);(5,9,10,22)(6,13,14,21)(7,17,18,31)(8,19,20,32)(11,24,15,28)(12,25,26,33)(16,29,30,36)(23,34,27,35)Polycyclic presentation:
2^9|a^2=h,c^2=fh,d^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,f^b=fi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72265848, #72265849, #72320041, … |
| maximal normal subgroup of | #72265848, #72265849, #72320041, … |
| maximal quotient of | #72265999 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 281255);