Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #479356 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17201 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
713 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,10,5,11,6,20)(3,12,16,24,15,8,13,22)(4,9,21,25,18,26,19,27)(14,23,29,32,17,28,30,31);(1,3,5,15)(2,8)(4,17,18,14)(6,13,7,16)(9,23)(10,22)(11,12)(19,29,21,30)(20,24)(25,32)(26,28)(27,31);(3,13)(6,7)(8,22)(10,20)(12,24)(14,30)(15,16)(17,29)(19,21)(23,32)(25,27)(28,31);(1,4,5,18)(2,9)(3,14,15,17)(6,21,7,19)(8,23)(10,27)(11,26)(12,28)(13,29,16,30)(20,25)(22,31)(24,32)Polycyclic presentation:
2^9|a^2=fg,b^2=ef,d^2=ef,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,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 | #72319592, #72319982, #72320075, … |
| maximal normal subgroup of | #72319592, #72319982, #72320075, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 386552);