Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #478423 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17147 |
| 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 | |
1006 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)(3,13,17,24,15,8,16,25)(4,18,20,27,21,9,22,28)(14,29,31,32,30,23,19,26);(1,3)(2,8,11,13)(4,19)(5,17)(6,16)(7,15)(9,26,18,32)(10,25,12,24)(14,20)(21,31)(22,30)(23,28,29,27);(1,4)(2,9)(3,14)(5,22)(6,20)(7,21)(8,23)(10,27)(11,18)(12,28)(13,29)(15,30)(16,31)(17,19)(24,26)(25,32);(2,10)(5,6)(8,24)(9,27)(11,12)(13,25)(14,30)(16,17)(18,28)(20,22)(23,32)(26,29)Polycyclic presentation:
2^9|a^2=fg,b^2=ef,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=def,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,g^c=gi,g^d=gi,h^d=hi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72251783, #72251791, #72251803, … |
| maximal normal subgroup of | #72251783, #72251791, #72251803, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 385619);