Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418533 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15263 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1642 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,8)(3,13)(4,18)(5,6)(10,19)(11,12)(14,23)(15,27)(16,17)(22,29)(25,26)(30,31);(1,2)(3,14)(4,19)(5,11)(6,8)(7,12)(9,16)(10,20)(13,24)(15,29)(17,23)(18,25)(21,26)(22,31)(27,32)(28,30);(1,3)(2,9)(4,15)(5,16)(6,13)(7,17)(8,14)(10,22)(11,23)(12,24)(18,31)(19,28)(20,30)(21,27)(25,32)(26,29);(1,4,5,20,7,18,6,21)(2,10,8,19,12,26,11,25)(3,15,16,30,17,31,13,27)(9,22,14,28,24,29,23,32)Polycyclic presentation:
2^9|d^2=e,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,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,f^c=fi,g^c=gi,h^c=hi,f^d=fi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 325729);