Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #478528 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17152 |
| 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 | |
727 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)(3,10,16,8)(4,17,20,29)(5,12)(6,13)(7,11)(9,19,26,22)(14,25)(15,28,21,24)(18,27)(23,30)(31,32);(1,3,5,15,7,16,6,21)(2,8,12,24,11,10,13,28)(4,18,19,14,20,30,22,31)(9,25,17,23,26,32,29,27);(1,4)(2,9,11,26)(3,14)(5,19)(6,22)(7,20)(8,23,10,27)(12,17,13,29)(15,30)(16,31)(18,21)(24,32,28,25);(1,3)(2,10,11,8)(4,14)(5,21)(6,15)(7,16)(9,23,26,27)(12,24,13,28)(17,25,29,32)(18,19)(20,31)(22,30)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=e,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=def,g^b=gi,h^b=hi,d^c=de,g^c=gi,f^d=fi,h^d=hi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72237068, #72237078, #72237084, … |
| maximal normal subgroup of | #72237068, #72237078, #72237084, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 385724);