Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #441638 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
937 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,10,7,11,6,12)(3,13,16,23,15,8,17,24)(4,9,20,28,19,27,21,26)(14,22,18,25,30,29,31,32);(1,3,5,16,7,15,6,17)(2,8,12,23,11,13,10,24)(4,18,20,30,19,31,21,14)(9,25,26,22,27,32,28,29);(1,4)(2,9,11,27)(3,14)(5,21)(6,20)(7,19)(8,22,13,29)(10,26,12,28)(15,30)(16,31)(17,18)(23,25,24,32);(2,10)(3,15)(5,6)(8,23)(9,26)(11,12)(13,24)(14,30)(20,21)(22,25)(27,28)(29,32)Polycyclic presentation:
2^9|a^2=g,b^2=efg,c^2=ef,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,e^b=ei,f^b=fi,e^c=ei,f^c=fi,g^c=gi,g^d=gi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72251784, #72251791, #72251797, … |
| maximal normal subgroup of | #72251784, #72251791, #72251797, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 348834);