Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437753 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1484 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)(3,6)(4,5)(7,8)(17,18,23,27)(19,28,29,32)(20,30,24,25)(21,26,22,31);(9,10)(11,15)(12,13)(14,16)(17,19,22,30,23,29,21,25)(18,24,26,28,27,20,31,32);(2,5)(6,8)(10,13)(14,15)(17,20)(18,25)(19,26)(21,28)(22,32)(23,24)(27,30)(29,31);(9,11,12,16)(10,14,13,15)(19,29)(20,24)(25,30)(28,32);(1,3,4,7)(2,6,5,8)(17,21,23,22)(18,26,27,31)(19,30,29,25)(20,28,24,32)Polycyclic presentation:
2^9|a^2=i,b^2=f,d^2=hi,e^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^b=ei,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6371901);