Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25446 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2428 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14100 |
| prime radical | #60469 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1422 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,14,8,12,6,13)(3,9,5,11,16,24,21,29)(4,10,20,27,19,28,22,30)(15,31,18,25,32,23,17,26);(1,3,7,21,8,16,6,5)(2,9,13,11,12,24,14,29)(4,17,22,15,19,18,20,32)(10,25,27,31,28,26,30,23);(1,4)(2,10)(3,15)(5,18)(6,22)(7,20)(8,19)(9,23)(11,26)(12,28)(13,30)(14,27)(16,32)(17,21)(24,31)(25,29);(1,5,6,16,8,21,7,3)(2,11,14,9,12,29,13,24)(4,18,22,32,19,17,20,15)(10,26,27,23,28,25,30,31)Polycyclic presentation:
2^9|a^2=h,b^2=eh,d^2=efg,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,e^b=ei,f^b=fi,h^b=hi,e^c=ei,g^c=gi,e^d=ei,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72215906, #72219685, #72219690, … |
| maximal normal subgroup of | #72215906, #72219685, #72219690, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 279977);