Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35346 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C215 |
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 | |
22962 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,3,4)(5,6)(7,8);(2,4)(9,10)(11,17)(12,16)(13,15)(14,18)(19,23)(20,22)(21,24);(6,8)(9,11)(10,14)(12,19)(13,18)(15,17)(16,21)(20,24)(22,23);(11,18)(14,17)(19,24)(21,23);(10,15)(14,17)(16,22)(21,23);(9,12,13,20)(10,16,15,22)(11,19,18,24)(14,21,17,23)Polycyclic presentation:
2^9|a^2=g,f^2=i,b^a=bg,c^a=ch,c^b=ci,e^b=ei,d^c=di128 conjugacy classes — view the full character table.
| maximal subgroup of | #72226740, #72239322, #72239341, … |
| maximal normal subgroup of | #72226740, #72239322, #72239341, … |
| maximal quotient of | #72239470, #72258186, #72264096, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7536317);