Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator | C22 | C25×C4 |
| Frattini, socle, cosocle | C23 | C26 |
| prime radical | C2×#3460 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7276 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2,4,3)(7,8)(10,12)(13,14)(15,21)(16,20)(17,19)(18,22)(23,27)(24,26)(25,28);(5,6)(7,10)(8,12)(9,11)(13,15)(14,18)(16,23)(17,22)(19,21)(20,25)(24,28)(26,27);(1,3,4,2)(5,7,9,8)(6,10,11,12)(15,22)(18,21)(23,28)(25,27);(6,11)(10,12)(14,19)(18,21)(20,26)(25,27);(5,8,9,7)(6,12,11,10);(13,16,17,24)(14,20,19,26)(15,23,22,28)(18,25,21,27)Polycyclic presentation:
2^9|a^2=i,c^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=cg,d^b=dh152 conjugacy classes — view the full character table.
| maximal subgroup of | #72238254, #72238255, #72238660, … |
| maximal normal subgroup of | #72238254, #72238255, #72238660, … |
| maximal quotient of | #72240774, #72263739 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7533250);