Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476484 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17090 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23400 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1044 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,12,7,10,5,11)(3,13,16,8,17,25,15,24)(4,18,21,26,22,27,20,9)(14,29,31,32,19,23,30,28);(1,3)(2,8)(4,19)(5,16)(6,15)(7,17)(9,23)(10,24)(11,25)(12,13)(14,22)(18,32)(20,30)(21,31)(26,29)(27,28);(1,4,5,21,7,22,6,20)(2,9,11,18,10,26,12,27)(3,14,15,31,17,19,16,30)(8,23,13,28,24,29,25,32);(2,10)(4,20)(5,6)(8,24)(9,18)(14,30)(15,16)(19,31)(21,22)(23,28)(26,27)(29,32)Polycyclic presentation:
2^9|a^2=f,c^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=def,e^b=ei,f^b=fi,h^b=hi,d^c=de,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72237064, #72237068, #72237082, … |
| maximal normal subgroup of | #72237064, #72237068, #72237082, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 383680);