Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3858029 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #16249 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3146 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(6,9)(7,11)(13,14,18,20)(15,17,16,19)(22,26)(24,27);(2,4)(5,6,8,9)(7,12,11,10)(13,15,18,16)(14,19,20,17)(21,22)(23,27)(24,28)(25,26);(1,2,3,4)(7,11)(10,12)(14,20)(17,19)(22,26)(24,27);(5,6)(7,12)(8,9)(10,11)(13,16,18,15)(14,17,20,19)(21,23,25,28)(22,24,26,27);(5,7)(6,10)(8,11)(9,12)(13,17)(14,16)(15,20)(18,19)(21,24)(22,23)(25,27)(26,28)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=fhi,d^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,e^b=ef,e^c=ef,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3937428);