Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| 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 | #25132 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1496 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)(3,7)(4,6)(5,8)(18,23)(20,28)(21,30)(24,27)(25,31)(29,32);(1,3,4,8)(2,5,6,7)(9,10)(11,13)(12,14)(15,16)(17,18,21,23,22,25,30,31)(19,20,27,28,26,29,24,32);(2,6)(3,8)(9,11,12,15)(10,13,14,16)(17,19)(18,20)(21,27)(22,26)(23,28)(24,30)(25,29)(31,32);(2,6)(5,7)(10,14)(13,16)(19,26)(20,29)(24,27)(28,32);(2,6)(5,7)(17,20,21,28,22,29,30,32)(18,24,23,19,25,27,31,26)Polycyclic presentation:
2^9|b^2=f,c^2=h,e^2=fg,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^b=ei,d^c=di,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7507677);