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 | |
11412 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2)(3,4)(5,6,7,8)(9,10)(11,12)(13,16)(14,17)(15,18)(19,20)(21,23)(22,24);(5,6)(7,8)(9,10)(11,12)(13,16)(14,17)(15,18)(19,20)(21,23)(22,24);(1,2,3,4)(9,11,15,13)(10,16,18,12)(14,19,22,21)(17,23,24,20);(1,2)(3,4)(9,12,15,16)(10,11,18,13)(14,20,22,23)(17,19,24,21);(9,13,15,11)(10,16,18,12)(14,19,22,21)(17,20,24,23);(9,14,15,22)(10,17,18,24)(11,19,13,21)(12,20,16,23)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=i,e^2=i,f^2=i,b^a=bg,c^a=ch,c^b=ci,d^c=dh,f^e=fi113 conjugacy classes — view the full character table.
| maximal subgroup of | #72234472, #72234475, #72234479, … |
| maximal normal subgroup of | #72234472, #72234475, #72234479, … |
| maximal quotient of | #72261486, #72261490, #72261514, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7594248);