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 | |
11450 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(2,4)(6,7)(10,15)(13,16)(17,20)(22,24);(5,6,8,7)(9,10)(11,13)(12,20)(14,15)(16,19)(17,23)(18,24)(21,22);(1,2)(3,4)(5,7,8,6)(9,11,14,19)(10,16,15,13)(12,18,23,21)(17,24,20,22);(9,11)(10,13)(12,21)(14,19)(15,16)(17,24)(18,23)(20,22);(9,12)(10,17)(11,18)(13,22)(14,23)(15,20)(16,24)(19,21);(9,13)(10,11)(12,22)(14,16)(15,19)(17,18)(20,21)(23,24)Polycyclic presentation:
2^9|b^2=gi,c^2=g,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=di,e^d=ei128 conjugacy classes — view the full character table.
| maximal subgroup of | #72313512, #72313631 |
| maximal normal subgroup of | #72313512, #72313631 |
| maximal quotient of | #72316142 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7538230);