Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31282 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2684 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14513 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1047 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)(3,15,19,28)(4,21,17,10)(5,24)(6,12)(7,14)(8,13)(9,18,29,20)(11,26)(16,32)(22,27)(23,31,25,30);(1,3)(2,9)(4,22)(5,25)(6,20)(7,19)(8,18)(10,24)(11,31)(12,28)(13,15)(14,29)(16,23)(17,26)(21,32)(27,30);(1,4,7,17)(2,10)(3,16,19,5)(6,23,8,25)(9,27)(11,29)(12,30)(13,31)(14,21)(15,24)(18,22,20,26)(28,32);(1,5,7,16)(2,11)(3,17,19,4)(6,22,8,26)(9,21)(10,29)(12,24)(13,32)(14,27)(15,30)(18,23,20,25)(28,31)Polycyclic presentation:
2^9|a^2=h,c^2=efg,d^2=efg,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72280281, #72284737, #72284759, … |
| maximal normal subgroup of | #72280281, #72284737, #72284759, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 295274);