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 |
| prime radical | #58991 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6581 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(3,7)(6,8)(9,10)(11,15)(12,16)(13,14)(17,18,21,24)(19,22,20,23);(1,2)(3,6)(4,5)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,22)(20,23)(21,24);(2,5)(6,8)(10,14)(11,16);(9,10)(11,12)(13,14)(15,16);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,16)(14,15)(17,19,21,20)(18,22,24,23);(1,3)(2,6)(4,7)(5,8)(9,12)(10,11)(13,15)(14,16)(17,20,21,19)(18,23,24,22)Polycyclic presentation:
2^9|a^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72241084, #72241105, #72241107, … |
| maximal normal subgroup of | #72241084, #72241105, #72241107, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7615566);