Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23872 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| 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 | |
3864 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)(7,17)(8,22)(9,15)(10,14)(11,16)(12,19)(13,23)(18,27)(20,29)(21,28)(24,31)(25,30)(26,34)(32,33)(35,36);(5,7,11,21)(6,12,16,29)(8,18,24,34)(9,20,10,19)(13,27,30,26)(14,28,15,17)(22,32,31,36)(23,35,25,33);(5,8)(6,13)(7,18)(9,23)(10,25)(11,24)(12,26)(14,22)(15,31)(16,30)(17,32)(19,33)(20,35)(21,34)(27,29)(28,36);(5,8)(6,13)(7,18)(9,25)(10,23)(11,24)(12,27)(14,31)(15,22)(16,30)(17,32)(19,35)(20,33)(21,34)(26,29)(28,36)Polycyclic presentation:
2^9|a^2=h,b^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,f^c=fi,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72261075, #72261238, #72261337, … |
| maximal normal subgroup of | #72261075, #72261238, #72261337, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266867);