Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806903 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
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 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2247 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)(3,4)(5,6)(7,10)(8,12)(9,11)(13,14)(15,25)(16,23)(17,28)(18,24)(19,22)(20,27)(21,26);(5,7,10,6)(8,12,9,11)(13,15)(14,20)(16,19)(17,26)(18,27)(21,22)(23,28)(24,25);(2,4)(13,16,17,27,18,26,19,15)(14,21,28,25,24,23,22,20);(5,8,10,9)(6,11,7,12)(13,17,18,19)(14,22,24,28)(15,26,27,16)(20,21,25,23);(5,9,10,8)(6,11,7,12)(13,16,19,15,18,26,17,27)(14,23,22,25,24,21,28,20)Polycyclic presentation:
2^9|b^2=fi,c^2=hi,d^2=f,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6718762);