Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #427666 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2527 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15693 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
656 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)(3,12,14,7)(4,16,17,8)(5,9)(6,10)(11,20)(13,24)(15,23,19,22)(18,26,29,25)(21,27)(28,32)(30,31);(1,3)(2,7)(4,13,17,27)(5,19)(6,14)(8,24,16,21)(9,23)(10,12)(11,22)(15,20)(18,30,29,28)(25,31,26,32);(1,4)(2,8)(3,13,14,27)(5,18)(6,17)(7,21,12,24)(9,25)(10,16)(11,26)(15,28,19,30)(20,29)(22,31,23,32);(2,9,10,11)(7,22,12,23)(8,25,16,26)(21,31,24,32)Polycyclic presentation:
2^9|a^2=eg,b^2=fg,c^2=ef,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,g^c=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72285150, #72286206, #72286218 |
| maximal normal subgroup of | #72285150, #72286206, #72286218 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 348944);