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 | |
11554 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)(9,10,15,17)(11,14,19,16)(12,13,18,24)(20,22,23,21);(5,6)(7,8)(9,11)(10,16)(12,20)(13,21)(14,17)(15,19)(18,23)(22,24);(2,4)(10,17)(11,19)(12,18)(21,22);(1,2)(3,4)(9,12,15,18)(10,13,17,24)(11,20,19,23)(14,22,16,21);(9,13)(10,18)(11,21)(12,17)(14,20)(15,24)(16,23)(19,22);(9,14)(10,19)(11,17)(12,22)(13,23)(15,16)(18,21)(20,24)Polycyclic presentation:
2^9|a^2=g,d^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 | #72238142, #72238219, #72238237, … |
| maximal normal subgroup of | #72238142, #72238219, #72238237, … |
| maximal quotient of | #72285105, #72297240, #72297251, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7593998);