Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417508 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | #35348 | C2 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C26 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
2587 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,3,4)(9,10)(11,19,22,15)(12,16,13,17)(14,18)(20,23)(21,24);(5,6)(7,8)(9,11)(10,15)(12,20)(13,21)(14,22)(16,24)(17,23)(18,19);(2,4)(6,8)(9,12,14,13)(10,16)(11,20,22,21)(15,23)(17,18)(19,24);(9,13,14,12)(10,17)(11,21,22,20)(15,24)(16,18)(19,23)Polycyclic presentation:
2^9|a^2=ef,c^2=ei,d^2=ei,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,h^b=hi,f^c=fi,h^c=hi,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72262070, #72262091 |
| maximal normal subgroup of | #72262070, #72262091 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 327688);