Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #387433 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2372 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14486 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1053 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)(3,14)(4,20,5,10)(6,12)(7,13)(8,17)(9,22,31,24)(11,25)(15,32,16,26)(18,29)(19,28)(21,27,23,30);(1,3)(2,8)(4,21)(5,23)(6,18)(7,19)(9,30)(10,32)(11,28)(12,14)(13,29)(15,22)(16,24)(17,25)(20,26)(27,31);(1,4,7,5)(2,9)(3,15)(6,22,25,24)(8,26,29,32)(10,11)(12,20)(13,31)(14,27,28,30)(16,19)(17,21)(18,23);(1,5,7,4)(2,10,13,20)(3,16)(6,24,25,22)(8,27)(9,11,31,12)(14,32)(15,19)(17,23)(18,21)(26,28)(29,30)Polycyclic presentation:
2^9|a^2=h,c^2=ef,d^2=fg,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^c=hi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72284489, #72284504, #72284607, … |
| maximal normal subgroup of | #72284489, #72284504, #72284607, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 294821);