Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7238 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(2,4)(5,6)(7,11)(8,9)(10,12)(14,19)(15,22)(16,24)(25,26);(1,2)(3,4)(13,14,18,19)(15,20,22,17)(16,21,24,27)(23,25,28,26);(6,9)(10,11)(13,15,18,22)(14,17,19,20)(16,23,24,28)(21,26,27,25);(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,23)(17,19)(18,22)(21,25)(24,28)(26,27);(5,7,8,12)(6,10,9,11)(13,16)(14,21)(15,23)(17,26)(18,24)(19,27)(20,25)(22,28);(13,17,18,20)(14,15,19,22)(16,25,24,26)(21,28,27,23)Polycyclic presentation:
2^9|b^2=i,c^2=i,e^2=hi,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7837710);