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 | |
7195 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)(8,10)(9,12)(13,14)(15,21)(16,18)(17,20)(19,22)(23,27)(24,25)(26,28);(1,2)(3,4)(13,14,18,16)(15,19,22,21)(17,24,25,20)(23,27,28,26);(5,6,8,10)(7,9,12,11)(14,16)(15,22)(20,24)(23,28);(5,6)(7,9)(8,10)(11,12)(13,15)(14,19)(16,21)(17,23)(18,22)(20,26)(24,27)(25,28);(5,7,8,12)(6,9,10,11)(13,16,18,14)(15,21,22,19)(17,20,25,24)(23,26,28,27);(13,17,18,25)(14,20,16,24)(15,23,22,28)(19,26,21,27)Polycyclic presentation:
2^9|b^2=i,c^2=hi,e^2=h,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7893435);