Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #467938 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16595 |
| 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 | |
723 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)(3,6)(4,13)(5,19,22,10)(7,11)(8,12)(9,16)(14,23,28,21)(15,29,31,27)(17,26)(18,25)(20,32,30,24);(1,3,7,12,8,11,6,2)(4,9,17,25,18,26,16,13)(5,14,21,19,22,28,23,10)(15,24,20,27,31,32,30,29);(1,4,7,16,8,18,6,17)(2,9,3,13,12,26,11,25)(5,20,21,15,22,30,23,31)(10,27,14,24,19,29,28,32);(1,5,7,21,8,22,6,23)(2,10,3,14,12,19,11,28)(4,15,16,30,18,31,17,20)(9,24,13,29,26,32,25,27)Polycyclic presentation:
2^9|a^2=eg,b^2=g,c^2=f,d^2=f,e^2=i,f^2=i,g^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,h^b=hi,d^c=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 375134);