Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #647656 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3925 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(9,10,13,14)(11,12,15,16)(17,18)(19,23)(20,24)(21,22);(6,8)(10,14)(12,16)(17,18,21,22)(19,23,24,20);(5,6,7,8)(17,19)(18,20)(21,24)(22,23);(1,2)(3,4)(9,11,13,15)(10,12,14,16)(17,20)(18,19)(21,23)(22,24);(9,12)(10,15)(11,14)(13,16)(17,18)(19,20)(21,22)(23,24)Polycyclic presentation:
2^9|a^2=fg,b^2=g,c^2=gh,d^2=fg,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef104 conjugacy classes — view the full character table.
| maximal subgroup of | #72256749, #72256751, #72256756, … |
| maximal normal subgroup of | #72256749, #72256751, #72256756, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 556520);