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 | |
12104 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(2,4)(6,8)(9,10,14,19)(11,15,21,22)(12,17,23,16)(13,18,24,20);(5,6,7,8)(9,11)(10,15)(12,18)(13,16)(14,21)(17,24)(19,22)(20,23);(1,2)(3,4)(9,12)(10,16)(11,20)(13,22)(14,23)(15,24)(17,19)(18,21);(9,12)(10,17)(11,18)(13,22)(14,23)(15,24)(16,19)(20,21);(9,11)(10,15)(12,20)(13,17)(14,21)(16,24)(18,23)(19,22);(9,13)(10,18)(11,17)(12,22)(14,24)(15,23)(16,21)(19,20)Polycyclic presentation:
2^9|a^2=i,b^2=g,b^a=bg,c^a=ch,c^b=ci,f^b=fi,e^d=ei122 conjugacy classes — view the full character table.
| maximal subgroup of | #72234473, #72234474, #72234480, … |
| maximal normal subgroup of | #72234473, #72234474, #72234480, … |
| maximal quotient of | #72240160, #72261686, #72261746, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7538778);