Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996335 |
| 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 ◁ C4×#124 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1856 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,10)(8,11)(9,12)(13,14,19,16,18,21,17,22)(15,23,24,20,26,28,25,27);(5,6)(7,10)(8,11)(9,12)(13,15)(14,20)(16,23)(17,24)(18,26)(19,25)(21,27)(22,28);(1,2,3,4)(5,7,10,6)(8,11,9,12)(14,21)(16,22)(20,27)(23,28);(5,8,10,9)(6,11,7,12)(13,16,19,21,18,22,17,14)(15,20,24,28,26,27,25,23);(5,9,10,8)(6,11,7,12)(13,17,18,19)(14,22,21,16)(15,24,26,25)(20,28,27,23)Polycyclic presentation:
2^9|a^2=fi,c^2=gi,d^2=fhi,e^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7106129);