Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #440843 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15669 |
| 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 | |
1076 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)(3,14,17,8)(4,18)(5,12)(6,13)(7,11)(9,16,24,22)(10,20)(15,29,32,25)(19,27)(21,28)(23,31,26,30);(1,3,6,16,7,17,5,22)(2,8,13,9,11,14,12,24)(4,15,20,30,21,32,19,31)(10,26,28,29,27,23,18,25);(1,3)(2,9,11,24)(4,15)(5,16)(6,22)(7,17)(8,12,14,13)(10,25,27,29)(18,26,28,23)(19,30)(20,31)(21,32);(1,4)(2,10)(3,15)(5,19)(6,20)(7,21)(8,23)(9,25)(11,27)(12,18)(13,28)(14,26)(16,30)(17,32)(22,31)(24,29)Polycyclic presentation:
2^9|a^2=efg,b^2=eg,c^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,h^a=hi,c^b=cg,d^b=de,f^b=fi,h^b=hi,g^c=gi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72215936, #72215972, #72216029, … |
| maximal normal subgroup of | #72215936, #72215972, #72216029, … |
| maximal quotient of | C2×#440843, #72252123, #72255802, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 348039);