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 | |
11294 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(2,4)(6,8)(9,10)(11,18)(12,20)(13,16)(14,15)(17,22)(19,23)(21,24);(5,6,7,8)(9,11,15,13)(10,16,14,18)(12,19,22,21)(17,23,20,24);(1,2)(3,4)(11,13)(16,18)(19,21)(23,24);(9,12)(10,17)(11,19)(13,21)(14,20)(15,22)(16,23)(18,24);(2,4)(9,13,15,11)(10,16,14,18)(12,21,22,19)(17,23,20,24);(1,2)(3,4)(9,14,15,10)(11,18,13,16)(12,20,22,17)(19,24,21,23)Polycyclic presentation:
2^9|b^2=g,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fhi,c^b=ci,e^c=ehi,f^e=fhi110 conjugacy classes — view the full character table.
| maximal subgroup of | #72234472, #72234484, D4×#3449 |
| maximal normal subgroup of | #72234472, #72234484, D4×#3449 |
| maximal quotient of | #72256854, #72257027, #72264077, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7572755);