Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7563317 |
| 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 | |
2632 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)(3,15)(4,21)(5,24)(6,10)(7,23)(8,14)(9,18)(11,12)(13,26)(16,27)(17,28)(19,31)(20,30)(22,25)(29,32);(1,3,7,18,8,20,26,19)(2,9,13,15,14,31,23,30)(4,22,6,16,11,28,24,32)(5,25,12,29,10,17,21,27);(1,4,7,6,8,11,26,24)(2,10,13,21,14,5,23,12)(3,16,19,22,20,32,18,28)(9,27,30,17,31,29,15,25);(1,5,7,12,8,10,26,21)(2,11,13,24,14,4,23,6)(3,17,19,29,20,25,18,27)(9,28,30,16,31,22,15,32);(1,6,7,11,8,24,26,4)(2,12,13,10,14,21,23,5)(3,16,18,28,20,32,19,22)(9,29,15,17,31,27,30,25)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,e^c=ei,h^d=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72214632 |
| maximal normal subgroup of | #72214632 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7470513);