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) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #14885 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3190 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,9,7)(8,11,12,10)(13,14)(15,16)(17,18)(19,20)(21,22,24,26)(23,25,27,28);(1,2,3,4)(6,7)(8,12)(14,18)(16,19)(21,23,24,27)(22,25,26,28);(1,2)(3,4)(8,12)(10,11)(14,18)(16,19)(21,22)(23,25)(24,26)(27,28);(5,7,9,6)(8,10,12,11)(13,15,17,20)(14,19,18,16)(21,23)(22,25)(24,27)(26,28);(5,8,9,12)(6,10,7,11)(13,16,17,19)(14,15,18,20)(21,22)(23,25)(24,26)(27,28)Polycyclic presentation:
2^9|a^2=fg,b^2=ghi,d^2=f,e^2=f,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, 3783906);