Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499054 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#481 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
694 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,11,24,23,32,8,15,10,14,26,13,9,16,22,31)(7,17,34,27,36,29,19,33,18,12,25,30,20,35,21,28);(1,2)(3,4)(5,7)(6,12)(8,21)(9,20)(10,18)(11,25)(13,30)(14,17)(15,33)(16,29)(19,22)(23,36)(24,27)(26,34)(28,31)(32,35);(2,4)(6,13,14,24)(8,11)(9,23)(12,27,17,30)(15,16,31,32)(19,34)(20,36)(21,25)(22,26)(28,29,33,35);(7,18)(12,17)(19,21)(20,36)(25,34)(27,30)(28,33)(29,35)Polycyclic presentation:
2^9|a^2=fh,c^2=e,f^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,g^a=gi,c^b=cg,d^b=di,f^b=fi,f^c=fh,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250011, #72250413, #72250418, … |
| maximal normal subgroup of | #72250011, #72250413, #72250418, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406583);