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 | |
7058 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)(6,10)(8,11)(13,14)(15,16)(17,25)(18,19)(20,22)(21,26)(23,27)(24,28);(1,2,3,4)(5,6,9,10)(7,11,12,8)(14,19)(15,22)(21,25)(23,28);(6,10)(7,12)(13,15,18,22)(14,20,19,16)(17,23,26,28)(21,27,25,24);(13,16,18,20)(14,15,19,22)(17,24,26,27)(21,23,25,28);(5,7)(6,8)(9,12)(10,11)(13,15)(14,20)(16,19)(17,23)(18,22)(21,27)(24,25)(26,28);(5,8)(6,7)(9,11)(10,12)(13,17)(14,21)(15,23)(16,24)(18,26)(19,25)(20,27)(22,28)Polycyclic presentation:
2^9|b^2=gh,c^2=h,d^2=h,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7623048);