Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25196 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2424 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14004 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1122 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,11,8,3,6,12)(4,15,18,9,19,25,17,13)(5,20,21,10,22,26,23,14)(16,27,29,28,30,24,31,32);(1,3,8,2)(4,13)(5,14,22,10)(6,11,7,12)(9,19)(15,18)(16,27)(17,25)(20,23,26,21)(24,30)(28,29)(31,32);(1,4,7,18,8,19,6,17)(2,9,12,15,3,13,11,25)(5,16,21,29,22,30,23,31)(10,24,20,28,14,27,26,32);(1,5)(2,10)(3,14)(4,16)(6,23)(7,21)(8,22)(9,24)(11,26)(12,20)(13,27)(15,28)(17,31)(18,29)(19,30)(25,32)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,c^2=f,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,g^c=gi,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72280350 |
| maximal normal subgroup of | #72280350 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 274707);