Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #612424 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2537 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3754 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)(6,7)(10,13)(12,15)(17,18,21,19)(20,23,24,22);(9,10,14,13)(11,15,16,12)(18,19)(22,23);(5,6,8,7)(9,11,14,16)(10,15,13,12)(17,19,21,18)(20,23,24,22);(1,2)(3,4)(5,7,8,6)(9,12)(10,11)(13,16)(14,15);(9,13,14,10)(11,12,16,15)(17,20,21,24)(18,22,19,23)Polycyclic presentation:
2^9|a^2=h,b^2=fh,c^2=fg,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch116 conjugacy classes — view the full character table.
| maximal subgroup of | #72297592, #72297594, #72297597, … |
| maximal normal subgroup of | #72297592, #72297594, #72297597, … |
| maximal quotient of | #72298342, #72298386, #72298503, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 520161);