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 | |
11342 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,4,3)(6,8)(12,20)(16,21)(18,24)(22,23);(5,6,7,8)(10,14)(15,17)(16,21)(22,23);(2,3)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24);(1,3,4,2);(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23);(9,12,13,20)(10,16,14,21)(11,18,19,24)(15,22,17,23)Polycyclic presentation:
2^9|a^2=h,b^2=g,d^2=h,e^2=i,f^2=i,b^a=bg,c^a=ch,f^a=fi,c^b=ci,d^c=dh,e^c=ei116 conjugacy classes — view the full character table.
| maximal subgroup of | #72311366, #72311370, #72311396, … |
| maximal normal subgroup of | #72311366, #72311370, #72311396, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7579880);