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 | |
24354 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)(16,23)(18,24)(21,22);(5,6)(7,8)(9,10)(11,15)(12,16)(13,17)(14,19)(18,22)(20,23)(21,24);(1,2)(3,4)(5,6)(7,8)(9,11)(10,14)(12,18)(13,19)(15,17)(16,21)(20,24)(22,23);(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23);(9,10)(11,14)(12,16)(13,17)(15,19)(18,21)(20,23)(22,24);(9,12)(10,16)(11,18)(13,20)(14,21)(15,22)(17,23)(19,24)Polycyclic presentation:
2^9|b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^d=ei116 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7630668, #72234473, #72234478, … |
| maximal normal subgroup of | C2×#7630668, #72234473, #72234478, … |
| maximal quotient of | #72233920, #72233921, C2×#7630668, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7537864);