Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #463155 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16465 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
912 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,13,6,14,8,12)(3,9,23,25,17,27,18,26)(4,19,21,29,22,11,5,10)(15,28,20,24,16,30,31,32);(1,3)(2,9)(4,20)(5,16)(6,17)(7,18)(8,23)(10,28)(11,24)(12,25)(13,26)(14,27)(15,21)(19,32)(22,31)(29,30);(1,4,8,5,6,22,7,21)(2,10,12,11,14,29,13,19)(3,15,23,20,17,16,18,31)(9,24,25,30,27,32,26,28);(1,5,7,22,6,21,8,4)(2,11,12,10,14,19,13,29)(3,16,18,31,17,15,23,20)(9,24,25,28,27,32,26,30)Polycyclic presentation:
2^9|a^2=eg,c^2=ef,d^2=f,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 370351);