Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #385905 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14417 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60454 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1428 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,12,7,8,5,11)(3,13,14,9,17,24,16,22)(4,18,19,10,21,26,20,23)(15,29,28,27,30,25,31,32);(2,8)(3,14)(5,6)(9,24)(13,22)(15,28)(16,17)(18,26)(19,20)(25,27)(29,32)(30,31);(1,3,5,14,7,17,6,16)(2,9,11,24,8,22,12,13)(4,15,20,28,21,30,19,31)(10,25,18,32,23,29,26,27);(1,4)(2,10)(3,15)(5,20)(6,19)(7,21)(8,23)(9,25)(11,18)(12,26)(13,27)(14,28)(16,31)(17,30)(22,29)(24,32)Polycyclic presentation:
2^9|a^2=h,c^2=eg,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,f^c=fi,h^c=hi,f^d=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72215936, #72216166, #72216181, … |
| maximal normal subgroup of | #72215936, #72216166, #72216181, … |
| maximal quotient of | C2×#385907, #72255855, #72286824 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293103);