Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C212×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator | C22 | C25×C4 |
| Frattini, socle, cosocle | C23 | C26 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
8418 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,23)(17,20)(19,24)(21,27)(22,26)(25,28);(6,9)(10,12)(13,15,17,22)(14,18,20,26)(16,21,24,28)(19,25,23,27);(13,16)(14,19)(15,21)(17,24)(18,25)(20,23)(22,28)(26,27);(16,24)(19,23)(21,28)(25,27);(14,20)(18,26)(19,23)(25,27);(5,7,8,11)(6,10,9,12)Polycyclic presentation:
2^9|a^2=i,b^2=h,f^2=g,b^a=bg,c^a=ch,e^a=eh,d^c=dh176 conjugacy classes — view the full character table.
| maximal subgroup of | #72281894, #72281937, #72311906, … |
| maximal normal subgroup of | #72281894, #72281937, #72311906, … |
| maximal quotient of | #72316293 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532620);