Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #421528 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15293 |
| 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 | |
959 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,7)(3,11)(4,14)(5,17)(8,20)(9,15)(10,18)(19,22)(21,30)(23,28)(26,32)(27,31);(1,2)(3,8,11,22)(4,15)(5,18)(6,10)(7,17)(9,16)(12,26,24,30)(13,20,25,19)(14,28)(21,31,32,27)(23,29);(1,3,5,13,6,11,17,25)(2,8,7,19,10,22,18,20)(4,12,16,27,14,24,29,31)(9,21,15,26,23,32,28,30);(1,4,5,16,6,14,17,29)(2,9,7,15,10,23,18,28)(3,12,13,27,11,24,25,31)(8,21,19,26,22,32,20,30)Polycyclic presentation:
2^9|b^2=f,c^2=e,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,f^c=fi,g^c=gi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72339027, #72339031, #72339042, … |
| maximal normal subgroup of | #72339027, #72339031, #72339042, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 328724);