Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31414 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1327 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,11,16)(7,17)(8,21)(9,12,10,15)(13,25)(14,24)(18,26)(19,31,32,36)(20,29)(22,27)(23,30)(28,33,35,34);(1,2)(3,4)(6,12)(7,18)(8,22)(9,10)(13,17)(14,21)(15,16)(19,33)(23,24)(26,29)(28,35)(32,34);(2,4)(5,7,10,20,11,18,9,25)(6,13,12,26,16,29,15,17)(8,19,22,33,24,32,23,34)(14,28,27,36,21,35,30,31);(5,8)(6,14)(7,19)(9,23)(10,22)(11,24)(12,27)(13,28)(15,30)(16,21)(17,31)(18,32)(20,33)(25,34)(26,36)(29,35)Polycyclic presentation:
2^9|a^2=h,c^2=eh,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,e^b=ei,f^b=fi,e^c=ei,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72285470, #72285492, #72285620, … |
| maximal normal subgroup of | #72285470, #72285492, #72285620, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 295579);