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 | |
7262 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)(8,12)(10,11)(13,14)(15,20)(16,18)(17,25)(19,21)(22,26)(23,27)(24,28);(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,15,19,16)(14,20,21,18)(17,24,26,23)(22,27,25,28);(6,7)(10,11)(14,21)(15,16)(17,26)(27,28);(13,16,19,15)(14,18,21,20)(17,24,26,23)(22,27,25,28);(5,7,9,6)(8,11,12,10)(13,17)(14,22)(15,23)(16,24)(18,27)(19,26)(20,28)(21,25);(5,8)(6,10)(7,11)(9,12)(13,18)(14,15)(16,21)(17,27)(19,20)(22,23)(24,25)(26,28)Polycyclic presentation:
2^9|b^2=gh,d^2=h,e^2=hi,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, 7625462);