Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #470603 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16690 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
904 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,10,6,8,18,23)(3,7,14,20,13,22,11,25)(4,15,17,26,16,9,19,24)(12,28,27,31,30,21,29,32);(1,3)(2,7,8,22)(4,12)(5,11)(6,13)(9,21,15,28)(10,25,23,20)(14,18)(16,30)(17,29)(19,27)(24,31,26,32);(2,8)(3,11)(5,18)(7,20)(12,29)(13,14)(17,19)(21,31)(22,25)(24,26)(27,30)(28,32);(1,4)(2,9)(3,12)(5,19)(6,16)(7,21)(8,15)(10,26)(11,27)(13,30)(14,29)(17,18)(20,31)(22,28)(23,24)(25,32)Polycyclic presentation:
2^9|a^2=g,b^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,e^b=ei,g^b=gi,d^c=de,g^c=gi,f^d=fi,g^d=gi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 377799);