Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #420395 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15277 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60454 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1332 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,12,8,13,7,25)(3,14,5,24,17,11,16,9)(4,18,22,30,23,29,21,10)(15,26,19,28,31,32,20,27);(1,3)(2,9)(4,19)(5,7)(6,16)(8,17)(10,27)(11,12)(13,24)(14,25)(15,22)(18,32)(20,23)(21,31)(26,29)(28,30);(1,4,6,21,8,23,7,22)(2,10,12,29,13,30,25,18)(3,15,5,20,17,31,16,19)(9,26,14,27,24,32,11,28);(1,5,7,3,8,16,6,17)(2,11,25,24,13,14,12,9)(4,20,22,15,23,19,21,31)(10,28,18,32,30,27,29,26)Polycyclic presentation:
2^9|a^2=f,c^2=e,d^2=ei,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,f^c=fi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72338547, #72338551, #72338560 |
| maximal normal subgroup of | #72338547, #72338551, #72338560 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 327592);