Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #214672 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2308 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3500 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3842 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)(5,6,10,13,11,15,9,16)(7,17,20,28,21,12,19,26)(8,22,23,29,24,14,25,30)(18,27,32,36,33,31,34,35);(2,4)(5,7,11,21)(6,12,15,17)(8,18,24,33)(9,20,10,19)(13,28,16,26)(14,27,22,31)(23,34,25,32)(29,36,30,35);(6,13)(9,10)(12,26)(14,29)(15,16)(17,28)(19,20)(22,30)(23,25)(27,35)(31,36)(32,34);(2,4)(5,8)(6,14)(7,18)(9,23)(10,25)(11,24)(12,27)(13,29)(15,22)(16,30)(17,31)(19,32)(20,34)(21,33)(26,35)(28,36)Polycyclic presentation:
2^9|a^2=h,b^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,h^b=hi,f^c=fi,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72302970, #72303145, #72303186, … |
| maximal normal subgroup of | #72302970, #72303145, #72303186, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266803);