Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3858029 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2407 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3620 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)(11,16)(12,13)(14,15)(17,18,22,20)(19,24,21,23);(5,6,7,8)(9,11,14,12)(10,13,15,16)(17,19)(18,23)(20,24)(21,22);(5,6)(7,8)(10,15)(13,16)(19,21)(23,24);(1,2)(3,4)(9,12,14,11)(10,16,15,13)(17,20,22,18)(19,23,21,24);(9,13,14,16)(10,12,15,11)(17,21,22,19)(18,24,20,23)Polycyclic presentation:
2^9|a^2=fg,b^2=fh,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,e^b=ef,e^c=ef,e^d=ef86 conjugacy classes — view the full character table.
| maximal subgroup of | #72281213, #72281522, #72281771 |
| maximal normal subgroup of | #72281213, #72281522, #72281771 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3767272);