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 | |
11676 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)(3,4)(6,8)(9,10)(11,15)(12,16)(13,21)(14,18)(17,22)(19,24)(20,23);(5,6)(7,8)(9,11,14,12)(10,15,18,16)(13,19,22,20)(17,23,21,24);(2,4)(9,10,14,18)(11,16,12,15)(13,17,22,21)(19,24,20,23);(1,2)(3,4)(9,10)(11,15)(12,16)(13,17)(14,18)(19,23)(20,24)(21,22);(9,12,14,11)(10,16,18,15)(13,20,22,19)(17,24,21,23);(9,13)(10,17)(11,19)(12,20)(14,22)(15,23)(16,24)(18,21)Polycyclic presentation:
2^9|b^2=i,c^2=i,e^2=i,b^a=bg,c^a=ch,f^a=fi,c^b=ci,d^c=dh,e^c=ei116 conjugacy classes — view the full character table.
| maximal subgroup of | #72268516, #72283147, C2×#7672320, … |
| maximal normal subgroup of | #72268516, #72283147, C2×#7672320, … |
| maximal quotient of | #72268520, #72279775, #72280204, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7579516);