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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2039 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(15,21)(18,27)(22,28)(23,26);(2,3)(13,14,16,20)(15,22,21,28)(17,25,24,19)(18,26,27,23);(1,2,4,3)(6,9)(10,12)(13,15)(14,18)(16,21)(17,26)(19,28)(20,27)(22,25)(23,24);(1,3,4,2)(14,19)(17,24)(18,28)(20,25)(22,27)(23,26);(5,7,8,11)(6,10,9,12)(15,21)(18,27)(22,28)(23,26)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=f,d^2=f,e^2=gi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dhi,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, 6363794);