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 | |
7345 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(16,24)(20,27)(23,26)(25,28);(1,2)(3,4)(5,6,9,7)(8,11,12,10)(13,14,19,22)(15,21,18,17)(16,20,24,27)(23,28,26,25);(6,7)(10,11)(13,15,19,18)(14,17,22,21)(16,23,24,26)(20,25,27,28);(5,6,9,7)(8,10,12,11)(13,16)(14,20)(15,23)(17,25)(18,26)(19,24)(21,28)(22,27);(5,7,9,6)(8,10,12,11)(13,17)(14,15)(16,25)(18,22)(19,21)(20,23)(24,28)(26,27);(5,8,9,12)(6,10,7,11)(13,18,19,15)(14,21,22,17)(16,26,24,23)(20,28,27,25)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=hi,d^2=h,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, 7893518);