Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6371551 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C211 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×#386 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | #62852 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
6111 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,11,7,10,21,3)(4,14,17,25,16,8,20,12)(5,9,19,24,18,23,30,13)(15,27,29,32,28,22,31,26);(1,3,6,10,7,11,21,2)(4,12,20,14,16,25,17,8)(5,13,19,23,18,24,30,9)(15,26,31,27,28,32,29,22);(1,4)(2,8)(3,12)(5,15)(6,20)(7,16)(9,22)(10,14)(11,25)(13,26)(17,21)(18,28)(19,31)(23,27)(24,32)(29,30);(1,5)(2,9)(3,13)(4,15)(6,19)(7,18)(8,22)(10,23)(11,24)(12,26)(14,27)(16,28)(17,29)(20,31)(21,30)(25,32);(5,18)(9,23)(13,24)(15,28)(19,30)(22,27)(26,32)(29,31)Polycyclic presentation:
2^9|a^2=h,b^2=fh,h^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,f^b=fi,g^b=gi,h^b=hi,f^c=fi,h^c=hi,e^d=ei113 conjugacy classes — view the full character table.
| maximal subgroup of | #72215901, #72215931, #72215938, … |
| maximal normal subgroup of | #72215901, #72215931, #72215938, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6278752);