Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #845297 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #14715 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4607 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(9,10,12,14)(11,15,16,13)(18,23)(21,24);(6,8)(9,11,12,16)(10,13,14,15)(17,18,22,23)(19,24,20,21);(5,6,7,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,20)(18,24,23,21);(1,2)(3,4)(9,11)(10,13)(12,16)(14,15)(17,20,22,19)(18,21,23,24);(9,10)(11,13)(12,14)(15,16)(17,21)(18,19)(20,23)(22,24)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=fh,d^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg92 conjugacy classes — view the full character table.
| maximal subgroup of | #72323327, #72323338, #72323370, … |
| maximal normal subgroup of | #72323327, #72323338, #72323370, … |
| maximal quotient of | #72323629, #72330194 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 754000);