Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1258513 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4229 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(6,7)(13,14)(15,20)(16,25)(17,22)(18,19)(21,26)(23,27)(24,28);(9,10,11,12)(14,19)(15,23)(21,25)(22,24);(1,2,4,3)(5,6,8,7)(9,10)(11,12)(13,15,18,23)(14,20,19,27)(16,24,26,22)(17,25,28,21);(5,6,8,7)(13,16,18,26)(14,21,19,25)(15,24,23,22)(17,27,28,20);(1,3,4,2)(5,7,8,6)(13,17)(14,22)(15,25)(16,20)(18,28)(19,24)(21,23)(26,27)Polycyclic presentation:
2^9|b^2=fh,c^2=fg,d^2=i,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef86 conjugacy classes — view the full character table.
| maximal subgroup of | #72280694, #72280735 |
| maximal normal subgroup of | #72280694, #72280735 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1201419);