Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #35563 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3418 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,12)(8,10)(9,11)(13,14,20,22)(15,18,16,19)(17,21,27,25)(23,26,24,28);(1,2,3,4)(13,15,19,14,20,16,18,22)(17,25,26,24,27,21,28,23);(5,7,9,8)(6,10,11,12)(13,16)(14,19)(15,20)(17,24)(18,22)(21,28)(23,27)(25,26);(5,8,9,7)(6,10,11,12)(13,17)(14,21)(15,23)(16,24)(18,26)(19,28)(20,27)(22,25);(13,18,20,19)(14,15,22,16)(17,26,27,28)(21,23,25,24)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=g,d^2=g,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,h^a=hi,c^b=ch,d^b=dh,e^c=ei,f^c=fi,h^c=hi,f^d=fi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6359393);