Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23886 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2361 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
978 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(6,11,14,22)(7,15,18,23)(8,19,21,20)(27,35,36,32);(1,3,4,2)(5,6,10,14)(7,16,18,25)(8,13,21,30)(9,22,24,11)(12,23,28,15)(17,32,34,35)(19,26,20,29)(27,33,36,31);(2,3)(5,7)(6,12)(8,17)(9,23)(10,18)(11,25)(13,27)(14,28)(15,24)(16,22)(19,31)(20,33)(21,34)(26,35)(29,32)(30,36);(5,8,9,20,10,21,24,19)(6,13,22,29,14,30,11,26)(7,17,23,33,18,34,15,31)(12,27,16,32,28,36,25,35)Polycyclic presentation:
2^9|a^2=h,b^2=ef,d^2=fh,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 285151);