Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468068 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16600 |
| 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 | |
1128 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)(3,11)(4,15)(5,19)(6,10)(7,13)(8,18)(9,20)(12,28)(14,27)(16,24)(17,25)(21,30)(22,32)(23,29)(26,31);(1,3)(2,7,10,22)(4,12)(5,13)(6,14)(8,23,15,21)(9,11,19,27)(16,29)(17,30)(18,31)(20,32)(24,28,25,26);(1,4)(2,8,10,15)(3,12)(5,16)(6,18)(7,21,22,23)(9,24,19,25)(11,26,27,28)(13,29)(14,31)(17,20)(30,32);(4,16)(5,20)(8,24)(9,19)(11,27)(12,29)(13,32)(15,25)(17,18)(21,26)(23,28)(30,31)Polycyclic presentation:
2^9|b^2=g,c^2=g,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,h^b=hi,d^c=de,e^c=ei,f^c=fi,h^c=hi,e^d=ei53 conjugacy classes — view the full character table.
| maximal subgroup of | #72215972, #72216130, #72216142, … |
| maximal normal subgroup of | #72215972, #72216130, #72216142, … |
| maximal quotient of | C2×#468068, #72252097, #72313994 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 375264);