Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24985 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #555 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2563 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1380 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6,11,16)(7,17)(8,21,24,27)(9,12,10,15)(13,20)(14,23,30,22)(18,29)(19,33)(25,26)(28,36)(31,32)(34,35);(1,2,4,3)(6,12,16,15)(7,18,20,25)(13,17)(14,27,30,21)(19,32,34,36)(26,29)(28,31)(33,35);(5,7)(6,13)(8,19)(9,18)(10,25)(11,20)(12,26)(14,28)(15,29)(16,17)(21,35)(22,32)(23,36)(24,34)(27,33)(30,31);(1,3,4,2)(5,8,9,22,11,24,10,23)(6,14,15,21,16,30,12,27)(7,19,18,32,20,34,25,36)(13,28,29,35,17,31,26,33)Polycyclic presentation:
2^9|a^2=h,b^2=eg,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 269240);