Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533195 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#548 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1860 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(13,14,15,20,17,22,18,19)(16,21,23,27,25,28,24,26);(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,15,17,18)(14,19,22,20)(16,24,25,23)(21,27,28,26);(1,3,4,2)(6,9)(10,12)(14,20)(15,18)(19,22)(21,27)(23,24)(26,28);(5,7)(6,10)(8,11)(9,12)(13,16,17,25)(14,21,22,28)(15,23,18,24)(19,26,20,27);(7,11)(10,12)(16,25)(21,28)(23,24)(26,27)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=fi,d^2=i,g^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,g^c=gi,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6459715);