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 | |
7272 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)(7,11)(10,12)(13,14)(15,19)(16,20)(17,24)(18,25)(21,26)(22,28)(23,27);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(14,15)(20,23)(21,24)(22,25);(6,9)(10,12)(13,15,19,14)(16,20,27,23)(17,21,26,24)(18,25,28,22);(13,16,19,27)(14,20,15,23)(17,18,26,28)(21,22,24,25);(5,6)(7,10)(8,9)(11,12)(13,17,19,26)(14,21,15,24)(16,18,27,28)(20,22,23,25);(5,7)(6,10)(8,11)(9,12)(13,18)(14,22)(15,25)(16,26)(17,27)(19,28)(20,24)(21,23)Polycyclic presentation:
2^9|b^2=gh,c^2=h,d^2=h,e^2=h,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7615986);