Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #58990 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5488 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,5,8)(3,7,4,6)(10,13)(14,16)(19,23)(22,24);(1,3,5,4)(2,6,8,7)(17,18)(19,22)(20,21)(23,24);(1,3,5,4)(2,7,8,6)(9,10)(11,14)(12,13)(15,16)(18,21)(19,23);(1,3,5,4)(2,7,8,6)(10,13)(14,16);(17,19,20,23)(18,22,21,24);(1,4,5,3)(2,6,8,7)(9,11,12,15)(10,14,13,16)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=g,d^2=g,e^2=i,f^2=gh,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^c=dh,e^c=ei128 conjugacy classes — view the full character table.
| maximal subgroup of | #72340537, #72340544, #72340574 |
| maximal normal subgroup of | #72340537, #72340544, #72340574 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7574210);