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 | |
11834 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,4,3)(6,8)(13,21)(16,20)(18,24)(22,23);(5,6)(7,8)(9,10,14,12)(11,15,19,17)(13,16,21,20)(18,22,24,23);(2,3)(10,12)(15,17)(16,20)(22,23);(1,3,4,2)(9,11,14,19)(10,15,12,17)(13,18,21,24)(16,22,20,23);(9,12,14,10)(11,17,19,15)(13,20,21,16)(18,23,24,22);(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)Polycyclic presentation:
2^9|a^2=h,b^2=i,d^2=hi,e^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 | #72311691, #72313036, C2×#7672588, … |
| maximal normal subgroup of | #72311691, #72313036, C2×#7672588, … |
| maximal quotient of | C2×#7672588, #72333459, #72333473, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7579784);