Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
8626 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)(5,6)(7,11)(8,9)(10,12)(13,14)(15,22)(16,26)(17,20)(18,21)(19,24)(23,27)(25,28);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,19)(16,26)(17,20)(18,21)(22,24)(23,28)(25,27);(6,9)(10,11)(13,15)(14,19)(16,23)(17,25)(18,24)(20,27)(21,22)(26,28);(5,7,8,12)(6,10,9,11)(13,16,18,17)(14,20,21,26)(15,23,24,25)(19,27,22,28);(15,24)(19,22)(23,25)(27,28);(13,17,18,16)(14,20,21,26)(15,25,24,23)(19,27,22,28)Polycyclic presentation:
2^9|b^2=g,d^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dh,c^b=ci,d^b=dhi,e^c=ehi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72261189, #72261361, #72279324, … |
| maximal normal subgroup of | #72261189, #72261361, #72279324, … |
| maximal quotient of | #72280397, #72280420, #72280482, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7548969);