Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3858029 |
| automorphism | Aut(G) order 1048576 |
| 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 | C2×#2671 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2828 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(6,9)(7,11)(13,14,16,19)(15,18,20,17)(21,22)(23,25)(24,27)(26,28);(2,4)(5,6,8,9)(7,12,11,10)(13,15,16,20)(14,17,19,18)(21,22,24,27)(23,25,28,26);(1,2)(3,4)(7,11)(10,12)(13,14)(15,17)(16,19)(18,20)(21,23,24,28)(22,25,27,26);(5,6)(7,12)(8,9)(10,11)(13,14)(15,17)(16,19)(18,20)(21,22)(23,25)(24,27)(26,28);(5,7)(6,10)(8,11)(9,12)(13,15)(14,18)(16,20)(17,19)(21,23)(22,26)(24,28)(25,27)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=fi,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, 3814543);