Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
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 | |
11116 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)(6,8)(10,15)(11,20)(18,24)(19,21);(5,6)(7,8)(9,10,14,15)(11,17,20,16)(12,18,22,24)(13,19,23,21);(2,4)(9,11,14,20)(10,16,15,17)(12,21,22,19)(13,18,23,24);(1,2)(3,4)(9,11)(10,17)(12,21)(13,18)(14,20)(15,16)(19,22)(23,24);(9,12)(10,18)(11,21)(13,17)(14,22)(15,24)(16,23)(19,20);(9,13)(10,19)(11,18)(12,16)(14,23)(15,21)(17,22)(20,24)Polycyclic presentation:
2^9|b^2=i,c^2=i,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^c=dh,f^e=fi107 conjugacy classes — view the full character table.
| maximal subgroup of | #72234477, #72234843 |
| maximal normal subgroup of | #72234477, #72234843 |
| maximal quotient of | #72281032, #72284114, #72284179, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7604685);