Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533195 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3340 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)(13,14,18,19,17,22,15,20)(16,21,24,26,25,28,23,27);(5,6,10,9)(7,12,8,11)(14,19)(15,18)(20,22)(21,26)(23,24)(27,28);(2,4)(5,7,10,8)(6,11,9,12)(13,15,17,18)(14,20,22,19)(16,24,25,23)(21,26,28,27);(5,8,10,7)(6,11,9,12)(13,16,17,25)(14,21,22,28)(15,23,18,24)(19,26,20,27);(5,9,10,6)(7,12,8,11)(16,25)(21,28)(23,24)(26,27)Polycyclic presentation:
2^9|a^2=fi,b^2=hi,c^2=h,d^2=h,e^2=hi,f^2=i,b^a=bf,c^a=cg,c^b=ch,f^b=fi,d^c=di,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6480643);