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 | |
11228 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)(6,8)(13,21)(17,22)(18,24)(20,23);(5,6)(7,8)(9,10)(11,12)(13,17)(14,15)(16,19)(18,20)(21,22)(23,24);(2,4)(10,15)(12,16)(17,22)(20,23);(1,2)(3,4)(9,11)(10,16)(12,15)(13,18)(14,19)(17,23)(20,22)(21,24);(9,12,14,16)(10,11,15,19)(13,20,21,23)(17,18,22,24);(9,13)(10,17)(11,18)(12,20)(14,21)(15,22)(16,23)(19,24)Polycyclic presentation:
2^9|e^2=i,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 | #72316126, #72316203 |
| maximal normal subgroup of | #72316126, #72316203 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7580867);