Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| 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 | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2237 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(5,6)(7,10)(8,12)(9,11)(14,19)(15,20)(21,27)(23,24);(2,3)(5,7,9,8)(6,10,11,12)(13,14,17,19)(15,22,20,28)(16,24,25,23)(18,27,26,21);(1,2,4,3)(5,8,9,7)(6,10,11,12)(13,15)(14,16)(17,20)(18,24)(19,25)(21,28)(22,27)(23,26);(1,3,4,2)(13,16,17,25)(14,20,19,15)(18,28,26,22)(21,23,27,24);(14,19)(15,20)(21,27)(23,24)Polycyclic presentation:
2^9|a^2=fi,b^2=g,c^2=fg,d^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6354314);