Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1853 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)(9,10,12,13)(11,15,16,14)(18,22)(23,31)(24,25)(26,32);(1,3,4,8)(2,5,6,7)(10,13)(14,15)(17,18,20,22)(19,26,27,32)(21,24,29,25)(23,30,31,28);(9,11,12,16)(10,14,13,15)(17,19)(18,23)(20,27)(21,30)(22,31)(24,26)(25,32)(28,29);(18,24)(21,29)(22,25)(23,26)(28,30)(31,32);(18,22)(19,27)(24,25)(28,30)Polycyclic presentation:
2^9|a^2=fi,b^2=fg,c^2=g,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72285096 |
| maximal normal subgroup of | #72285096 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6353997);