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 | |
11616 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)(11,16,12,18)(13,14,23,24)(19,20,21,22);(5,6,7,8)(9,11)(10,16)(12,15)(13,19)(14,20)(17,18)(21,23)(22,24);(2,3)(10,17)(11,12)(13,23)(20,22);(1,3,4,2)(9,12,15,11)(10,18,17,16)(13,21,23,19)(14,22,24,20);(9,13)(10,14)(11,19)(12,21)(15,23)(16,20)(17,24)(18,22);(9,14)(10,13)(11,20)(12,22)(15,24)(16,19)(17,23)(18,21)Polycyclic presentation:
2^9|a^2=h,b^2=g,d^2=h,b^a=bg,c^a=ch,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72262026, #72262034, #72262211, … |
| maximal normal subgroup of | #72262026, #72262034, #72262211, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7580064);