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 | |
11379 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,4,3)(6,8)(9,10,15,17)(11,18,12,16)(13,19,23,22)(14,20,24,21);(5,6,7,8)(9,11,15,12)(10,16,17,18)(13,21,23,20)(14,19,24,22);(2,3)(10,17)(11,12)(19,22)(20,21);(1,3,4,2)(9,12,15,11)(10,18,17,16)(13,20,23,21)(14,22,24,19);(9,13)(10,19)(11,21)(12,20)(14,18)(15,23)(16,24)(17,22);(9,14)(10,20)(11,19)(12,22)(13,16)(15,24)(17,21)(18,23)Polycyclic presentation:
2^9|a^2=h,b^2=g,d^2=h,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^c=dh,f^e=fi107 conjugacy classes — view the full character table.
| maximal subgroup of | #72234477, #72234482, #72234484, … |
| maximal normal subgroup of | #72234477, #72234482, #72234484, … |
| maximal quotient of | #72261754, #72261768, #72284012, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7602273);