Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438975 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2127 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)(8,9)(11,12)(14,20)(16,27)(21,28)(22,23);(2,4)(13,14,18,20)(15,22,25,23)(16,19,27,17)(21,24,28,26);(6,9)(10,12)(13,15,19,26,18,25,17,24)(14,21,16,22,20,28,27,23);(13,16,19,20,18,27,17,14)(15,23,26,21,25,22,24,28);(5,7,8,11)(6,10,9,12)(13,17,18,19)(14,16,20,27)(15,24,25,26)(21,22,28,23)Polycyclic presentation:
2^9|b^2=i,c^2=h,d^2=h,e^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=dh,h^b=hi,d^c=di,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6374507);