Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9519 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
844 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6)(7,13,15,10)(8,11)(9,12)(14,17,20,19)(16,18)(22,27)(23,31)(24,25)(28,35)(32,33)(34,36);(5,7)(6,10)(8,14)(9,15)(11,17)(12,13)(16,20)(18,19)(21,22,25,30,26,27,24,29)(23,28,32,35,33,34,31,36);(1,2,3,4)(5,8,9,16)(6,11,12,18)(7,14,15,20)(10,17,13,19)(21,23,24,31,26,33,25,32)(22,28,29,36,27,34,30,35)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=efh,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gi,d^b=di,e^b=eh,f^b=fh,d^c=dh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 17981);