Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #466064 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23402 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1076 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)(3,4)(5,21)(6,13)(7,12)(8,14)(9,16)(10,19)(11,23)(15,30)(17,20)(18,31)(22,27)(24,29)(25,28)(26,32);(1,3)(2,9,14,19)(4,13,20,12)(5,15)(6,16)(7,10)(8,17)(11,22,29,30)(18,21,32,28)(23,26)(24,31)(25,27);(1,4,6,19,8,20,7,9)(2,10,12,3,14,16,13,17)(5,22,23,32,25,30,24,18)(11,27,21,31,29,15,28,26);(1,5,6,24,8,25,7,23)(2,11,12,28,14,29,13,21)(3,15,16,31,17,27,10,26)(4,18,19,30,20,32,9,22)Polycyclic presentation:
2^9|b^2=fg,c^2=e,d^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,h^b=hi,d^c=de,g^c=gi,h^c=hi,e^d=ei53 conjugacy classes — view the full character table.
| maximal subgroup of | #72287341, #72287345, #72287354, … |
| maximal normal subgroup of | #72287341, #72287345, #72287354, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 373260);