Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32696 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2749 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
975 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,11,14,9,16,10,15)(7,12,20,29,18,28,19,27)(8,21,24,31,23,30,25,13)(17,32,33,36,22,35,34,26);(1,2)(3,4)(5,7,9,18)(6,12,16,28)(8,22,23,17)(10,20,11,19)(13,26,31,36)(14,27,15,29)(21,35,30,32)(24,33,25,34);(2,4)(5,8,11,25,9,23,10,24)(6,13,15,21,16,31,14,30)(7,17,20,34,18,22,19,33)(12,26,27,32,28,36,29,35);(6,14,16,15)(7,18)(12,27,28,29)(13,30,31,21)(17,22)(19,20)(26,32,36,35)(33,34)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=ef,d^2=ei,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^b=hi,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 291625);