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 | C2×#3035 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2378 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6,8,9)(7,10,11,12)(13,14,18,19)(15,23,26,20)(16,21,25,28)(17,22,24,27);(6,9)(10,12)(13,15)(14,20)(16,27)(17,28)(18,26)(19,23)(21,24)(22,25);(2,4)(13,16,18,25)(14,21,19,28)(15,24,26,17)(20,22,23,27);(5,7,8,11)(6,10,9,12)(13,17,18,24)(14,22,19,27)(15,25,26,16)(20,21,23,28);(13,14,18,19)(15,20,26,23)(16,28,25,21)(17,22,24,27)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,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, 6346653);