Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455826 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2631 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2002 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1331 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(6,10)(7,14)(8,16)(11,20)(12,18)(13,19);(1,2)(3,4)(5,6,8,10,9,13,16,19)(7,12,17,18,14,11,15,20);(2,4)(5,7,8,15,9,14,16,17)(6,11,10,18,13,12,19,20);(5,7)(6,12)(8,17)(9,14)(10,18)(11,13)(15,16)(19,20)Polycyclic presentation:
2^9|b^2=ef,c^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,g^b=gi,h^b=hi,d^c=de,f^c=fi,e^d=ei,f^d=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72338546, #72338548, #72338559 |
| maximal normal subgroup of | #72338546, #72338548, #72338559 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 363154);