Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500661 |
| 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 | #34911 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1665 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)(4,7)(5,8)(10,13)(11,15);(1,3,6,2)(4,8,5,7)(9,10,12,13)(11,14,15,16)(17,18)(19,27)(20,24)(21,25)(22,26)(23,28)(29,31)(30,32);(1,4,6,5)(2,7,3,8)(9,11,12,15)(10,14,13,16)(17,19)(18,23)(20,28)(21,30)(22,29)(24,31)(25,27)(26,32);(2,3)(7,8)(10,13)(14,16)(17,20,22,21)(18,24,26,25)(19,28,29,30)(23,31,32,27);(1,5,6,4)(2,8,3,7)(17,20,22,21)(18,25,26,24)(19,28,29,30)(23,27,32,31)Polycyclic presentation:
2^9|b^2=f,c^2=f,d^2=i,e^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,e^b=ei,h^b=hi,h^c=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72257267, #72257411, #72264267 |
| maximal normal subgroup of | #72257267, #72257411, #72264267 |
| maximal quotient of | #72265097 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7420472);