Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #427874 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15375 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2008 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1246 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,8)(3,12)(4,17)(5,7)(9,22)(10,26)(11,21)(13,25)(14,29)(19,20)(27,31)(28,32);(1,2)(3,13)(4,18)(5,21)(6,11)(7,8)(9,16)(10,17)(12,22)(14,30)(15,25)(19,26)(20,23)(24,27)(28,29)(31,32);(1,3)(2,9)(4,14)(5,15)(6,12)(7,16)(8,22)(10,24)(11,25)(13,21)(17,29)(18,28)(19,31)(20,27)(23,32)(26,30);(1,4,7,17,6,20,5,19)(2,10,8,23,11,26,21,18)(3,14,16,29,12,27,15,31)(9,24,22,32,25,30,13,28)Polycyclic presentation:
2^9|d^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,g^c=gi,h^c=hi,e^d=ei,f^d=fi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72237063, #72237069, #72237090, … |
| maximal normal subgroup of | #72237063, #72237069, #72237090, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 335070);