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 | |
7162 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)(14,19)(15,23)(17,27)(24,26);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,14,18,19)(15,21,23,20)(16,17,22,27)(24,25,26,28);(6,9)(10,12)(13,15,18,23)(14,20,19,21)(16,26,22,24)(17,25,27,28);(13,16,18,22)(14,17,19,27)(15,24,23,26)(20,28,21,25);(5,6)(7,10)(8,9)(11,12)(13,15)(14,21)(16,24)(17,25)(18,23)(19,20)(22,26)(27,28);(5,7)(6,10)(8,11)(9,12)(13,17)(14,22)(15,25)(16,19)(18,27)(20,24)(21,26)(23,28)Polycyclic presentation:
2^9|b^2=g,c^2=h,d^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, 7615685);