Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #458129 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2420 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2018 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1080 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,15,11,16,9,13)(7,17,20,28,21,27,18,12)(8,22,24,14,25,29,23,30)(19,26,34,31,33,36,32,35);(1,2,3,4)(5,7,10,20,11,21,9,18)(6,12,13,27,16,28,15,17)(8,19,24,34,25,33,23,32)(14,26,22,35,30,36,29,31);(2,4)(6,13)(7,18)(8,23)(9,10)(15,16)(17,27)(19,33)(20,21)(22,29)(24,25)(26,31)(35,36);(5,8,10,24,11,25,9,23)(6,14,15,29,16,30,13,22)(7,19,20,34,21,33,18,32)(12,26,17,31,28,36,27,35)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 373048);