Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #456514 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23409 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1148 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,11,26,10,16,9,15)(7,12,19,13,20,29,18,28)(8,21,24,14,23,33,25,32)(17,31,34,36,22,30,35,27);(1,2)(3,4)(5,7,9,18,10,20,11,19)(6,12,15,28,16,29,26,13)(8,17,24,34,23,22,25,35)(14,27,33,31,32,36,21,30);(2,4)(5,7)(6,13)(8,22)(9,19)(10,20)(11,18)(12,26)(14,31)(15,29)(16,28)(17,23)(21,36)(24,34)(25,35)(27,33)(30,32);(5,8,9,24,10,23,11,25)(6,14,15,33,16,32,26,21)(7,17,18,34,20,22,19,35)(12,27,28,31,29,36,13,30)Polycyclic presentation:
2^9|a^2=efi,b^2=f,d^2=f,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,h^b=hi,d^c=de,f^c=fi,g^c=gi,e^d=ei,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 366602);