Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24472 |
| 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 | |
2136 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,13,8,14,7,12)(3,15,18,29,19,32,17,9)(4,20,16,28,23,31,22,10)(5,24,25,30,26,27,21,11);(1,3)(2,9)(4,21)(5,16)(6,18)(7,17)(8,19)(10,30)(11,28)(12,32)(13,15)(14,29)(20,27)(22,26)(23,25)(24,31);(1,4,7,22,8,23,6,16)(2,10,13,20,14,28,12,31)(3,5,17,21,19,26,18,25)(9,27,15,11,29,24,32,30);(1,5)(2,11)(3,16)(4,17)(6,25)(7,21)(8,26)(9,28)(10,29)(12,27)(13,24)(14,30)(15,31)(18,23)(19,22)(20,32)Polycyclic presentation:
2^9|a^2=h,c^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,g^b=gi,g^c=gi,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72241162, #72241172, #72241302, … |
| maximal normal subgroup of | #72241162, #72241172, #72241302, … |
| maximal quotient of | C2×#360593 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 267789);