Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7563313 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #59126 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2623 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,17)(4,23)(5,6)(7,15)(8,14)(9,16)(10,21)(11,25)(12,13)(18,30)(19,24)(20,28)(22,32)(26,29)(27,31);(1,3,7,21,9,22,8,27)(2,10,14,17,16,31,15,32)(4,24,6,20,12,29,25,18)(5,26,23,28,11,19,13,30);(1,4,8,25,9,12,7,6)(2,11,15,23,16,5,14,13)(3,18,21,24,22,20,27,29)(10,28,17,19,31,30,32,26);(1,5,8,13,9,11,7,23)(2,12,15,6,16,4,14,25)(3,19,21,30,22,26,27,28)(10,29,17,18,31,24,32,20);(1,6,7,12,9,25,8,4)(2,13,14,5,16,23,15,11)(3,20,21,29,22,18,27,24)(10,30,17,26,31,28,32,19)Polycyclic presentation:
2^9|b^2=f,c^2=g,d^2=g,e^2=f,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72214352, #72214466, #72214632, … |
| maximal normal subgroup of | #72214352, #72214466, #72214632, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7470509);