Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #686200 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2410 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
5492 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(6,8)(9,10,13,14)(11,15,16,12)(20,24)(22,23);(2,4)(6,8)(9,11,13,16)(10,12,14,15)(17,18,21,19)(20,22,24,23);(1,2,3,4)(5,6,7,8)(10,14)(11,16)(17,19,21,18)(20,23,24,22);(9,10)(11,12)(13,14)(15,16)(17,20)(18,22)(19,23)(21,24);(9,12)(10,11)(13,15)(14,16)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=hi,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=df92 conjugacy classes — view the full character table.
| maximal subgroup of | #72264830, #72332470, #72332473, … |
| maximal normal subgroup of | #72264830, #72332470, #72332473, … |
| maximal quotient of | #72332976, #72332986, #72336263, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 620125);