Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62862 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C26 |
| commutator | C22 | C25×C4 |
| prime radical | C2×#3549 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7644 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2,4,3)(5,6)(7,11)(8,10)(9,12)(13,14,19,20)(15,16,17,21)(18,26,27,22)(23,28,25,24);(5,6,8,10)(7,9,12,11)(14,20)(15,17)(22,26)(23,25);(5,6)(7,9)(8,10)(11,12)(13,15,19,17)(14,21,20,16)(18,23,27,25)(22,28,26,24);(5,7)(6,9)(8,12)(10,11)(13,16,19,21)(14,15,20,17)(18,24,27,28)(22,23,26,25);(1,3,4,2)(13,17,19,15)(14,21,20,16)(18,25,27,23)(22,28,26,24);(13,18,19,27)(14,22,20,26)(15,23,17,25)(16,24,21,28)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=h,d^2=h,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=cg,e^b=eh,d^c=dh140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7533759);