Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C215 |
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 | |
22592 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)(5,6,7,8)(9,10)(11,20)(12,16)(13,17)(14,18)(15,19)(21,24)(22,23);(5,6)(7,8)(9,11,15,17)(10,13,19,20)(12,22,14,21)(16,24,18,23);(1,2)(3,4)(5,6)(7,8)(9,12,15,14)(10,16,19,18)(11,21,17,22)(13,23,20,24);(9,10)(11,13)(12,16)(14,18)(15,19)(17,20)(21,23)(22,24);(9,13)(10,17)(11,19)(12,23)(14,24)(15,20)(16,22)(18,21);(9,14,15,12)(10,18,19,16)(11,22,17,21)(13,24,20,23)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=i,f^2=i,b^a=bg,c^a=ch,c^b=ci,f^b=fi,e^d=ei122 conjugacy classes — view the full character table.
| maximal subgroup of | #72262355, #72262356, #72313664, … |
| maximal normal subgroup of | #72262355, #72262356, #72313664, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7540485);