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 | |
7422 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)(3,4)(6,7)(10,11)(13,14)(15,22)(16,21)(17,18)(19,20)(23,28)(24,27)(25,26);(5,6,9,7)(8,10,12,11)(13,15)(14,18)(16,23)(17,19)(20,22)(21,26)(24,25)(27,28);(2,4)(5,7,9,6)(8,10,12,11)(14,20)(15,17)(16,24)(26,28);(1,2)(3,4)(5,8,9,12)(6,10,7,11)(13,16,19,24)(14,21,20,27)(15,23,17,25)(18,26,22,28);(13,17,19,15)(14,18,20,22)(16,25,24,23)(21,26,27,28);(13,18)(14,15)(16,26)(17,20)(19,22)(21,23)(24,28)(25,27)Polycyclic presentation:
2^9|b^2=gi,c^2=gi,d^2=g,e^2=i,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7585468);