Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #427842 |
| 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 | |
1254 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)(3,13)(4,18)(5,11)(6,10)(7,12)(8,14)(9,20)(15,23)(16,22)(17,24)(19,26)(21,25)(27,29)(28,31)(30,32);(3,14)(4,19)(5,6)(8,22)(9,25)(10,11)(13,24)(15,28)(16,17)(23,31)(27,32)(29,30);(1,3)(2,8)(4,15)(5,17)(6,14)(7,16)(9,23)(10,22)(11,13)(12,24)(18,27)(19,30)(20,28)(21,29)(25,32)(26,31);(1,4,5,21,7,19,6,20)(2,9,10,26,12,25,11,18)(3,15,17,29,16,30,14,28)(8,23,22,31,24,32,13,27)Polycyclic presentation:
2^9|d^2=ef,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72251785, #72251791, #72251799, … |
| maximal normal subgroup of | #72251785, #72251791, #72251799, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 335038);