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 | |
7876 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 28 points:
(2,4)(5,6)(7,11)(8,9)(10,12)(13,14)(15,21)(16,24)(17,19)(18,22)(20,26)(23,27)(25,28);(1,2)(3,4)(13,15)(14,18)(16,25)(17,22)(19,21)(20,28)(23,26)(24,27);(6,9)(10,11)(14,19)(15,22)(20,24)(23,25);(5,6)(7,10)(8,9)(11,12)(13,15)(14,18)(16,23)(17,22)(19,21)(20,27)(24,28)(25,26);(5,7,8,12)(6,10,9,11)(13,16,17,26)(14,20,19,24)(15,23,22,25)(18,27,21,28);(13,14)(15,18)(16,20)(17,19)(21,22)(23,27)(24,26)(25,28)Polycyclic presentation:
2^9|e^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,e^b=ei,d^c=dh,f^c=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72326930, #72327652, #72331611 |
| maximal normal subgroup of | #72326930, #72327652, #72331611 |
| maximal quotient of | #72327031, #72327911, #72328265, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7654501);