Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #458125 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #555 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2563 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
948 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,17)(8,22)(9,12)(10,15)(11,16)(13,20)(14,25)(18,30)(19,27)(21,28)(23,29)(24,32)(26,31)(33,35)(34,36);(6,12,16,15)(13,28,17,27)(14,22,32,31)(29,36,33,30);(5,7)(6,13,16,17)(8,23)(9,21)(10,19)(11,20)(12,27,15,28)(14,30,32,36)(18,24)(22,33,31,29)(25,34)(26,35);(1,2,3,4)(5,8,10,25,11,26,9,24)(6,14,12,22,16,32,15,31)(7,18,21,35,20,34,19,23)(13,29,28,36,17,33,27,30)Polycyclic presentation:
2^9|b^2=g,c^2=g,d^2=h,e^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,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 383229);