Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12715 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1781 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| socle | C2 | C2×#1781 |
| prime radical | #62812 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×#585 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
3339 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)(3,14,17,27)(4,10)(5,11,19,8)(6,12)(7,13)(9,15,26,21)(16,29)(18,24,20,23)(22,30)(25,32)(28,31);(2,8)(5,19)(9,22)(10,23)(11,13)(12,24)(14,28)(16,31)(18,20)(25,26)(27,29)(30,32);(1,3)(2,9)(4,15)(5,16)(6,21)(7,17)(8,22)(10,14)(11,25)(12,27)(13,26)(18,30)(19,31)(20,32)(23,28)(24,29);(1,4,7,6)(2,10,13,12)(3,15,17,21)(5,18,19,20)(8,23,11,24)(9,14,26,27)(16,30,31,32)(22,28,25,29)Polycyclic presentation:
2^9|a^2=g,d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=ei,g^b=ghi,f^c=fi,g^c=gi,g^e=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72236949, #72237540, #72237550, … |
| maximal normal subgroup of | #72236949, #72237540, #72237550, … |
| maximal quotient of | #72260426, #72261207, #72261210, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400615);