Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
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 | |
7198 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,4,7)(3,6,8,5)(10,14)(12,15)(17,18,20,22)(19,21,24,23);(3,8)(5,6)(9,10,13,14)(11,15,16,12);(1,3)(2,5)(4,8)(6,7)(10,14)(12,15)(17,18)(19,23)(20,22)(21,24);(3,8)(5,6)(9,11,13,16)(10,15,14,12)(17,19)(18,21)(20,24)(22,23);(1,3)(2,6)(4,8)(5,7)(9,11)(10,12)(13,16)(14,15);(1,2)(3,6)(4,7)(5,8)(9,12)(10,11)(13,15)(14,16)Polycyclic presentation:
2^9|a^2=h,b^2=g,d^2=g,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,f^c=fi,e^d=ei110 conjugacy classes — view the full character table.
| maximal subgroup of | #72241092, #72241121, #72241127, … |
| maximal normal subgroup of | #72241092, #72241121, #72241127, … |
| maximal quotient of | #72328603, #72328871, #72329122, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7589170);