Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #427241 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2471 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15669 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1087 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)(3,12,15,8)(4,16,19,28)(5,11)(6,21)(7,10)(9,18,27,22)(13,30)(14,25,20,24)(17,29)(23,31)(26,32);(1,3,6,14,7,15,5,20)(2,8,21,24,10,12,11,25)(4,17,18,32,19,31,22,13)(9,26,16,23,27,30,28,29);(1,4)(2,9,10,27)(3,13)(5,18)(6,22)(7,19)(8,23,12,29)(11,16,21,28)(14,31)(15,32)(17,20)(24,26,25,30);(8,12)(23,29)(24,25)(26,30)Polycyclic presentation:
2^9|a^2=efg,b^2=eg,c^2=e,f^2=i,g^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=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72215936, #72215972, #72216077, … |
| maximal normal subgroup of | #72215936, #72215972, #72216077, … |
| maximal quotient of | C2×#440842, #72252136, #72256002, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 348038);