Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4452694 |
| automorphism | Aut(G) order 4194304 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1990 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(8,12)(10,11)(16,23)(20,25)(21,27)(26,28)(29,30,33,36)(31,37,38,34)(32,35,40,42)(39,43,44,41);(2,3)(5,6,9,7)(8,10,12,11)(14,18)(19,24)(20,25)(26,28)(31,38)(34,37)(39,44)(41,43);(2,3)(5,7,9,6)(8,11,12,10)(13,14)(15,19)(16,20)(17,18)(21,26)(22,24)(23,25)(27,28)(29,30)(31,34)(32,35)(33,36)(37,38)(39,41)(40,42)(43,44);(1,3,4,2)(5,8)(6,10)(7,11)(9,12)(13,15,17,22)(14,19,18,24)(16,21,23,27)(20,26,25,28)(29,31)(30,34)(32,39)(33,38)(35,41)(36,37)(40,44)(42,43);(13,16)(14,20)(15,21)(17,23)(18,25)(19,26)(22,27)(24,28)(29,32,33,40)(30,35,36,42)(31,39,38,44)(34,41,37,43)Polycyclic presentation:
2^9|a^2=fg,b^2=gi,c^2=gi,d^2=fh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,d^c=df80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4363877);