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 | |
8919 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)(13,14)(15,20)(16,18)(17,26)(19,21)(22,24)(23,27)(25,28);(1,2,3,4)(5,6)(7,10)(8,11)(9,12)(13,15,19,23)(14,20,21,27)(16,24,25,17)(18,22,28,26);(5,7)(6,9)(8,12)(10,11)(14,21)(15,23)(16,25)(22,26);(13,16)(14,18)(15,24)(17,23)(19,25)(20,22)(21,28)(26,27);(5,7)(6,10)(8,12)(9,11)(13,17,19,24)(14,22,21,26)(15,16,23,25)(18,20,28,27);(5,6)(7,9)(8,11)(10,12)(13,18)(14,16)(15,22)(17,27)(19,28)(20,24)(21,25)(23,26)Polycyclic presentation:
2^9|b^2=gh,e^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi101 conjugacy classes — view the full character table.
| maximal subgroup of | #72284836, #72302940, #72302976, … |
| maximal normal subgroup of | #72284836, #72302940, #72302976, … |
| maximal quotient of | #72326461, #72326483, #72326700, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7757639);