Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7504292 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2684 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3021 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2280 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,8)(3,13)(4,18)(5,6)(10,19)(11,12)(15,27)(16,17)(22,25)(23,26)(29,32)(30,31);(1,2,6,8,7,12,5,11)(3,14,13,22,17,9,16,25)(4,19,20,26,18,23,21,10)(15,29,30,28,31,32,27,24);(1,3)(2,9)(4,15)(5,13)(6,16)(7,17)(8,22)(10,24)(11,25)(12,14)(18,31)(19,32)(20,27)(21,30)(23,29)(26,28);(1,4)(2,10)(3,15)(5,20)(6,21)(7,18)(8,23)(9,24)(11,19)(12,26)(13,27)(14,28)(16,30)(17,31)(22,29)(25,32);(1,5,7,6)(2,11,12,8)(3,13,17,16)(4,20,18,21)(9,25,14,22)(10,19,26,23)(15,27,31,30)(24,32,28,29)Polycyclic presentation:
2^9|b^2=g,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,f^a=fi,g^a=gi,c^b=ch,d^b=df,h^b=hi,g^c=gi,f^d=fi,g^d=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72215721, #72216203, #72216589, … |
| maximal normal subgroup of | #72215721, #72216203, #72216589, … |
| maximal quotient of | #72250782, C2×#7504294, #72255644, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7411490);