Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24420 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2410 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13877 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2113 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,11,8,12,6,16)(3,4,14,9,15,19,24,18)(5,20,21,10,23,26,22,27)(13,25,29,32,30,31,28,17);(1,3)(2,9)(4,16)(5,13)(6,24)(7,14)(8,15)(10,25)(11,19)(12,18)(17,20)(21,29)(22,28)(23,30)(26,32)(27,31);(1,4,6,18,8,19,7,9)(2,3,11,14,12,15,16,24)(5,17,22,31,23,32,21,25)(10,13,26,29,27,30,20,28);(1,5)(2,10)(3,13)(4,17)(6,22)(7,21)(8,23)(9,25)(11,26)(12,27)(14,29)(15,30)(16,20)(18,31)(19,32)(24,28)Polycyclic presentation:
2^9|a^2=h,c^2=f,e^2=i,f^2=i,h^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,g^c=gi,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72241168, #72241172, #72241305, … |
| maximal normal subgroup of | #72241168, #72241172, #72241305, … |
| maximal quotient of | C2×#360585 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 267781);