Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437418 |
| 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 | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3240 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,10)(8,11)(9,12)(13,14,18,22)(15,23,16,20)(17,19,26,21)(24,28,25,27);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(14,19)(17,26)(20,27)(21,22)(23,28)(24,25);(2,4)(5,7,9,8)(6,11,12,10)(13,15,18,16)(14,20,22,23)(17,24,26,25)(19,27,21,28);(2,4)(5,8,9,7)(6,11,12,10)(13,16,18,15)(14,20,22,23)(17,25,26,24)(19,27,21,28);(15,16)(20,23)(24,25)(27,28)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=g,d^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,e^c=ei,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6360118);