Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #441222 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15674 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60423 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1244 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)(3,13,18,8)(4,19,21,29)(5,11)(6,12)(7,10)(9,20,28,22)(14,24)(15,30)(16,26,17,25)(23,31)(27,32);(1,3,6,16,7,18,5,17)(2,8,12,25,10,13,11,26)(4,15,20,32,21,31,22,14)(9,27,19,23,28,24,29,30);(1,4)(2,9,10,28)(3,14)(5,20)(6,22)(7,21)(8,23,13,30)(11,19,12,29)(15,17)(16,31)(18,32)(24,25,27,26);(1,4)(2,9,10,28)(3,15)(5,22)(6,20)(7,21)(8,24,13,27)(11,29,12,19)(14,17)(16,32)(18,31)(23,25,30,26)Polycyclic presentation:
2^9|a^2=efg,b^2=eg,c^2=e,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^b=hi,g^c=gi,h^c=hi,f^d=fi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72216047, #72216077, #72216192, … |
| maximal normal subgroup of | #72216047, #72216077, #72216192, … |
| maximal quotient of | #72237182, C2×#441222, #72252211, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 348418);