Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #440901 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15670 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
964 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,11,5,10,6,20)(3,12,15,24,16,8,13,22)(4,17,21,30,18,9,19,26)(14,25,28,32,29,23,27,31);(1,3,6,15,5,16,7,13)(2,8,20,22,10,12,11,24)(4,14,19,28,18,29,21,27)(9,25,30,32,17,23,26,31);(3,13)(6,7)(8,22)(11,20)(12,24)(14,27)(15,16)(19,21)(23,31)(25,32)(26,30)(28,29);(1,4)(2,9,10,17)(3,14)(5,18)(6,19)(7,21)(8,23,12,25)(11,26,20,30)(13,27)(15,28)(16,29)(22,31,24,32)Polycyclic presentation:
2^9|a^2=fg,b^2=eg,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^b=hi,g^c=gi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72319815 |
| maximal normal subgroup of | #72319815 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 348097);