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 |
| prime radical | C2×#2464 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3793 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(6,7)(13,14,18,19)(15,20,23,26)(16,24,25,21)(17,22,27,28);(9,10,11,12)(14,19)(15,23)(21,24)(22,28);(1,2,4,3)(5,6,8,7)(9,10)(11,12)(13,15)(14,20)(16,22)(17,21)(18,23)(19,26)(24,27)(25,28);(5,6,8,7)(13,16,18,25)(14,21,19,24)(15,22,23,28)(17,26,27,20);(1,3,4,2)(5,7,8,6)(13,17)(14,22)(15,21)(16,20)(18,27)(19,28)(23,24)(25,26)Polycyclic presentation:
2^9|a^2=f,b^2=fh,c^2=g,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 | #72280790, #72280804 |
| maximal normal subgroup of | #72280790, #72280804 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1201416);