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 | |
10854 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)(13,21)(16,23)(18,24)(20,22);(5,6)(7,8)(9,10,14,17)(11,12,19,15)(13,16,21,23)(18,20,24,22);(2,4)(9,11,14,19)(10,15,17,12)(13,18,21,24)(16,22,23,20);(1,2)(3,4);(9,12,14,15)(10,11,17,19)(13,20,21,22)(16,18,23,24);(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)Polycyclic presentation:
2^9|b^2=i,c^2=i,e^2=i,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei110 conjugacy classes — view the full character table.
| maximal subgroup of | #72280094, #72302966, #72307705, … |
| maximal normal subgroup of | #72280094, #72302966, #72307705, … |
| maximal quotient of | #72281381, #72315727 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7585463);