Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9091 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#386 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2570 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)(7,17)(8,22,26,31)(9,14)(10,15)(11,16)(12,20)(13,24,32,25)(18,27,35,36)(19,29)(21,28)(23,33,34,30);(1,2)(3,4)(5,7)(6,12)(8,23)(9,19)(10,20)(11,21)(13,30)(14,17)(15,28)(16,29)(18,25)(22,27)(24,35)(26,34)(31,36)(32,33);(2,4)(5,8,9,24,11,26,10,25)(6,13,15,31,16,32,14,22)(7,18,20,34,21,35,19,23)(12,27,17,33,29,36,28,30);(5,9,11,10)(6,14,16,15)(7,19,21,20)(8,24,26,25)(12,17,29,28)(13,22,32,31)(18,23,35,34)(27,33,36,30)Polycyclic presentation:
2^9|a^2=h,c^2=e,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72237649, #72237653, #72237668, … |
| maximal normal subgroup of | #72237649, #72237653, #72237668, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265135);