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 | |
9002 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2,3,4)(5,6)(7,11)(8,9)(10,12)(13,14,18,20)(15,21,22,16)(17,19,26,28)(23,27,25,24);(1,2)(3,4)(13,15)(14,16)(17,25)(18,22)(19,27)(20,21)(23,26)(24,28);(6,9)(10,11)(15,22)(16,21)(23,25)(24,27);(5,6)(7,10)(8,9)(11,12)(13,15)(14,16)(17,23)(18,22)(19,24)(20,21)(25,26)(27,28);(5,7)(6,10)(8,12)(9,11)(13,16)(14,15)(17,24)(18,21)(19,23)(20,22)(25,28)(26,27);(13,17,18,26)(14,19,20,28)(15,23,22,25)(16,24,21,27)Polycyclic presentation:
2^9|a^2=g,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72256663, #72256676, #72326920, … |
| maximal normal subgroup of | #72256663, #72256676, #72326920, … |
| maximal quotient of | #72327021, #72327983, #72329164, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7650664);