Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470428 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1632 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,8)(4,6)(5,7)(11,15)(14,16)(17,18,22,20)(19,25,26,32)(21,29,30,24)(23,31,28,27);(9,10)(11,14)(12,13)(15,16)(17,19,21,27,22,26,30,31)(18,23,24,25,20,28,29,32);(1,3,5,4)(2,6,7,8)(10,13)(14,16)(18,20)(19,26)(24,29)(27,31);(1,4,5,3)(2,6,7,8)(10,13)(14,16)(17,20,22,18)(19,23,26,28)(21,29,30,24)(25,31,32,27);(9,11,12,15)(10,14,13,16)(18,20)(19,26)(24,29)(27,31)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=gi,d^2=g,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,e^b=ei,d^c=di,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6403181);