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×#3481 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7398 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)(5,6,9,10)(7,12,8,11)(13,14)(15,23)(16,20)(17,22)(18,21)(19,25)(24,28)(26,27);(6,10)(7,8)(13,15)(14,19)(16,26)(17,24)(18,25)(20,28)(21,23)(22,27);(5,7,9,8)(6,11,10,12)(13,16,18,17)(14,20,21,22)(15,24,25,26)(19,27,23,28);(1,3,4,2)(5,8,9,7)(6,12,10,11)(14,21)(19,23)(20,22)(27,28);(6,10)(11,12);(13,17,18,16)(14,22,21,20)(15,24,25,26)(19,27,23,28)Polycyclic presentation:
2^9|a^2=i,c^2=g,d^2=i,f^2=gh,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 | #72284610, #72284726, #72285285, … |
| maximal normal subgroup of | #72284610, #72284726, #72285285, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7533331);