Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2282 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)(5,6)(7,10)(8,12)(9,11)(13,14,17,18)(15,19,20,16)(23,27)(24,28);(5,7,9,12)(6,10,11,8)(13,14)(15,16)(17,18)(19,20)(21,22,25,26)(23,28,27,24);(5,8)(6,7)(9,10)(11,12)(13,15,17,20)(14,16,18,19)(21,23)(22,24)(25,27)(26,28);(1,2,3,4)(5,7)(6,8)(9,12)(10,11)(13,16,17,19)(14,15,18,20)(21,22)(23,24)(25,26)(27,28);(5,8)(6,7)(9,10)(11,12)(13,15)(14,16)(17,20)(18,19)(21,24)(22,23)(25,28)(26,27)Polycyclic presentation:
2^9|a^2=f,b^2=fg,c^2=f,d^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3375536);