Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #423848 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15316 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3068 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1301 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,10,7,11,6,23)(3,12,15,25,17,13,16,8)(4,18,20,9,22,19,21,26)(14,24,29,27,31,32,30,28);(1,2)(3,13)(4,19)(5,23)(6,10)(7,11)(8,16)(9,20)(12,17)(14,24)(15,25)(18,22)(21,26)(27,30)(28,29)(31,32);(1,3,6,16,7,17,5,15)(2,8,10,12,11,25,23,13)(4,14,21,30,22,31,20,29)(9,24,19,27,26,32,18,28);(1,4,5,20,7,22,6,21)(2,9,10,19,11,26,23,18)(3,14,15,29,17,31,16,30)(8,24,12,27,25,32,13,28)Polycyclic presentation:
2^9|a^2=e,c^2=f,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,g^c=gi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72213913, #72219485, #72219486 |
| maximal normal subgroup of | #72213913, #72219485, #72219486 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 331044);