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 | |
11918 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)(12,20)(16,21)(18,24)(22,23);(5,6,7,8)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24);(1,2)(3,4)(10,14)(15,17)(16,21)(22,23);(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24);(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23);(9,12,13,20)(10,16,14,21)(11,18,19,24)(15,22,17,23)Polycyclic presentation:
2^9|b^2=g,f^2=i,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=di116 conjugacy classes — view the full character table.
| maximal subgroup of | #72312244, #72312270, #72318091, … |
| maximal normal subgroup of | #72312244, #72312270, #72318091, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7537804);