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 | |
1041 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11,7,13,5,12)(3,14,16,8,17,26,23,27)(4,18,20,9,22,10,21,28)(15,24,30,29,31,32,19,25);(1,3)(2,8)(4,19)(5,23)(6,16)(7,17)(9,25)(10,24)(11,26)(12,14)(13,27)(15,20)(18,32)(21,31)(22,30)(28,29);(1,4)(2,9,13,28)(3,15,17,31)(5,20)(6,21)(7,22)(8,24)(10,11,18,12)(14,29)(16,19,23,30)(25,26)(27,32);(1,4)(2,10)(3,15,17,31)(5,20)(6,21)(7,22)(8,25,27,29)(9,11)(12,28)(13,18)(14,24,26,32)(16,19,23,30)Polycyclic presentation:
2^9|a^2=h,c^2=ef,d^2=fg,e^2=i,f^2=i,g^2=i,h^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 | #72339651, #72339655, #72339658, … |
| maximal normal subgroup of | #72339651, #72339655, #72339658, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 294822);