Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #686200 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2397 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
5754 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(6,7)(13,14,18,17)(15,20,23,19)(16,24,26,21)(22,27,25,28);(10,12)(13,15,18,23)(14,19,17,20)(16,22,26,25)(21,27,24,28);(5,6,8,7)(9,10)(11,12)(13,15)(14,20)(16,25)(17,19)(18,23)(21,27)(22,26)(24,28);(1,2,3,4)(5,7,8,6)(13,16,18,26)(14,21,17,24)(15,22,23,25)(19,27,20,28);(13,17,18,14)(15,19,23,20)(16,24,26,21)(22,27,25,28)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=g,d^2=i,e^2=f,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=df92 conjugacy classes — view the full character table.
| maximal subgroup of | #72280858 |
| maximal normal subgroup of | #72280858 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 626343);