Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C215 |
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 | |
23656 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);(5,6)(7,8)(9,10)(11,18)(12,15)(13,21)(14,17)(16,22)(19,20)(23,24);(1,2)(3,4)(5,6)(7,8)(9,11,14,15)(10,12,17,18)(13,19,22,23)(16,20,21,24);(9,10)(11,12)(13,16)(14,17)(15,18)(19,20)(21,22)(23,24);(9,12)(10,15)(11,17)(13,20)(14,18)(16,23)(19,21)(22,24);(9,13)(10,16)(11,19)(12,20)(14,22)(15,23)(17,21)(18,24)Polycyclic presentation:
2^9|c^2=i,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 | #72240149, #72241325, #72241326, … |
| maximal normal subgroup of | #72240149, #72241325, #72241326, … |
| maximal quotient of | #72240152, #72240156, #72240159, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7539034);