Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24472 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2410 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13975 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1438 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,10,7,11,5,21)(3,12,15,8,16,22,13,24)(4,17,18,9,20,25,19,26)(14,23,30,31,29,32,28,27);(1,3,6,15,7,16,5,13)(2,8,10,22,11,24,21,12)(4,14,18,30,20,29,19,28)(9,23,25,31,26,32,17,27);(3,13,16,15)(8,22,24,12)(14,28,29,30)(23,31,32,27);(1,4)(2,9)(3,14)(5,19)(6,18)(7,20)(8,23)(10,25)(11,26)(12,27)(13,28)(15,30)(16,29)(17,21)(22,31)(24,32)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=f,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72215974, #72216144, #72219684, … |
| maximal normal subgroup of | #72215974, #72216144, #72219684, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 272645);