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 | |
11160 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)(10,15)(11,19)(17,18)(20,21);(5,6)(7,8)(10,15)(16,22)(17,18)(23,24);(1,2)(3,4)(9,10)(11,16)(12,17)(13,18)(14,15)(19,22)(20,23)(21,24);(9,11)(10,16)(12,20)(13,21)(14,19)(15,22)(17,23)(18,24);(2,4)(9,12,14,13)(10,17,15,18)(11,20,19,21)(16,23,22,24);(1,2)(3,4)(9,13,14,12)(10,18,15,17)(11,21,19,20)(16,24,22,23)Polycyclic presentation:
2^9|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 | D4×#3449, #72257515 |
| maximal normal subgroup of | D4×#3449, #72257515 |
| maximal quotient of | #72257050, #72264057, #72264105, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7572754);