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 | |
7126 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(17,18)(19,23)(20,22)(21,24);(1,2,5,3)(4,6,8,7)(9,11,14,16)(10,15,12,13)(17,19)(18,21)(20,24)(22,23);(2,3)(6,7)(10,12)(11,16)(18,22)(21,23);(9,11,14,16)(10,15,12,13)(17,18)(19,21)(20,22)(23,24);(1,3,5,2)(4,7,8,6)(9,12,14,10)(11,13,16,15)(17,19)(18,21)(20,24)(22,23);(1,4)(2,6)(3,7)(5,8)(9,13)(10,11)(12,16)(14,15)Polycyclic presentation:
2^9|b^2=i,d^2=gh,e^2=i,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 | #72232390, #72233117, #72238571, … |
| maximal normal subgroup of | #72232390, #72233117, #72238571, … |
| maximal quotient of | #72232445, #72232478 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7618400);