Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #400057 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2631 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2008 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1704 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,11,17)(7,14,8,22)(9,24)(10,18,12,16)(13,21,29,20)(15,25)(19,33)(23,28)(26,32)(27,31)(30,35)(34,36);(1,2)(3,4)(5,7,12,21,11,8,10,20)(6,13,18,22,17,29,16,14)(9,19,25,34,27,23,26,35)(15,28,24,36,32,33,31,30);(2,4)(5,8)(6,14)(7,11)(9,23)(10,21)(12,20)(13,18)(15,30)(16,29)(17,22)(19,27)(24,28)(25,35)(26,34)(31,33)(32,36);(5,9,12,25,11,27,10,26)(6,15,18,24,17,32,16,31)(7,19,21,34,8,23,20,35)(13,28,22,36,29,33,14,30)Polycyclic presentation:
2^9|a^2=f,b^2=ef,d^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,g^b=gi,h^b=hi,g^c=gi,h^c=hi,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72285083, #72285452, #72285492, … |
| maximal normal subgroup of | #72285083, #72285452, #72285492, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 308202);