Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25196 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2424 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13899 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1416 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,8,4,15)(5,10,19,28)(6,12)(7,11)(9,18,25,14)(13,29)(16,22)(17,23)(20,26,21,27)(24,30)(31,32);(1,3,6,18,7,4,16,14)(2,8,22,9,11,15,12,25)(5,13,21,30,19,17,20,31)(10,23,26,24,28,29,27,32);(1,4)(2,9)(3,7)(5,17)(6,14)(8,12)(10,24)(11,25)(13,19)(15,22)(16,18)(20,30)(21,31)(23,27)(26,29)(28,32);(1,5,6,21,7,19,16,20)(2,10,22,26,11,28,12,27)(3,13,18,30,4,17,14,31)(8,23,9,24,15,29,25,32)Polycyclic presentation:
2^9|a^2=h,b^2=ef,d^2=ef,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72215931, #72216141, #72219685 |
| maximal normal subgroup of | #72215931, #72216141, #72219685 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 268685);