Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C23×#121 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
7694 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)(5,6,7,8)(9,10)(11,15)(12,13)(14,16)(17,18,21,19)(20,23,24,22);(10,13)(14,15)(18,19)(22,23);(1,2)(3,4)(5,6)(7,8)(17,18,21,19)(20,22,24,23);(1,2)(3,4)(5,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,21,18)(20,23,24,22);(9,11)(10,14)(12,16)(13,15)(17,20)(18,22)(19,23)(21,24)Polycyclic presentation:
2^9|a^2=i,c^2=h,d^2=h,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df116 conjugacy classes — view the full character table.
| maximal subgroup of | #72232039, #72232044, #72232045, … |
| maximal normal subgroup of | #72232039, #72232044, #72232045, … |
| maximal quotient of | #72323577, #72323584, #72323614, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 428851);