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 | |
2663 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,9,14)(3,15,19,17)(4,12,23,21)(5,7,13,8)(6,11,22,28)(10,24,26,18)(16,29,31,25)(20,27,32,30);(1,3,7,18,9,19,8,24)(2,10,5,15,14,26,13,17)(4,20,22,31,23,32,6,16)(11,27,21,29,28,30,12,25);(1,4,7,22,9,23,8,6)(2,11,5,21,14,28,13,12)(3,16,24,32,19,31,18,20)(10,25,17,30,26,29,15,27);(1,5,8,2,9,13,7,14)(3,17,18,10,19,15,24,26)(4,21,6,11,23,12,22,28)(16,30,20,25,31,27,32,29);(1,6,8,23,9,22,7,4)(2,12,13,28,14,21,5,11)(3,16,24,32,19,31,18,20)(10,25,17,30,26,29,15,27)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=gi,d^2=g,e^2=fi,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 | #72214383, #72214424, #72214632, … |
| maximal normal subgroup of | #72214383, #72214424, #72214632, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7470512);