Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667836 |
| 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 | #25168 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1710 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,14)(12,13)(15,16)(17,18,21,26)(19,27,29,23)(20,24,22,25)(28,31,30,32);(1,3)(2,5)(4,8)(6,7)(17,19)(18,23)(20,28)(21,29)(22,30)(24,32)(25,31)(26,27);(2,6)(5,7)(10,13)(14,16)(17,20)(18,24)(21,22)(25,26)(28,30)(31,32);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,20,21,22)(18,25,26,24)(19,28,29,30)(23,31,27,32);(3,8)(5,7)(17,20)(18,25)(19,29)(21,22)(24,26)(31,32)Polycyclic presentation:
2^9|a^2=f,d^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72239589, #72251178, #72251283 |
| maximal normal subgroup of | #72239589, #72251178, #72251283 |
| maximal quotient of | #72239748, #72264613 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6606976);