Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #422233 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2424 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3068 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1564 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)(7,17)(8,22)(9,16)(10,13)(11,15)(12,18)(14,25)(19,30)(20,29)(21,27)(23,28)(24,31)(26,32)(33,36)(34,35);(1,2)(3,4)(5,7)(6,12)(8,23)(9,20)(10,21)(11,18)(13,29)(14,30)(15,27)(16,17)(19,24)(22,35)(25,33)(26,34)(28,31)(32,36);(2,4)(6,13)(7,18,21,20)(12,27,29,17)(14,22)(15,16)(19,23,34,33)(28,35,36,30)(31,32);(5,8,9,24,10,25,11,26)(6,14,15,31,13,22,16,32)(7,19,18,23,21,34,20,33)(12,28,17,30,29,36,27,35)Polycyclic presentation:
2^9|c^2=fi,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 330822);