Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417617 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2678 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15277 |
| prime radical | #60444 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1357 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,11,7,13,6,12)(3,14,16,26,17,25,23,8)(4,18,21,10,22,28,20,9)(15,24,30,27,31,32,19,29);(1,3)(2,8)(4,19)(5,23)(6,16)(7,17)(9,27)(10,29)(11,25)(12,14)(13,26)(15,20)(18,24)(21,31)(22,30)(28,32);(1,4,5,20,7,22,6,21)(2,9,11,28,13,10,12,18)(3,15,16,19,17,31,23,30)(8,24,14,29,26,32,25,27);(1,4,5,20,7,22,6,21)(2,10,11,18,13,9,12,28)(3,15,16,19,17,31,23,30)(8,24,14,29,26,32,25,27)Polycyclic presentation:
2^9|a^2=f,c^2=e,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,f^c=fi,h^c=hi,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72216047, #72216059, #72216203, … |
| maximal normal subgroup of | #72216047, #72216059, #72216203, … |
| maximal quotient of | C2×#420400, #72252179, #72256247, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 327596);