Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2760365 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
1943 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6,8,10)(7,9,11,12)(13,14)(15,18)(16,21)(17,20)(19,24)(22,26)(23,28)(25,27)(29,30,34,32)(31,35,38,39)(33,40,41,36)(37,43,44,42);(5,7,8,11)(6,9,10,12)(14,15)(19,21)(20,22)(27,28)(30,32)(35,39)(36,40)(42,43);(5,7)(6,9)(8,11)(10,12)(13,15,18,14)(16,19,24,21)(17,22,26,20)(23,27,25,28)(29,31)(30,35)(32,39)(33,37)(34,38)(36,42)(40,43)(41,44);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,16,18,24)(14,19,15,21)(17,25,26,23)(20,28,22,27)(29,32,34,30)(31,35,38,39)(33,36,41,40)(37,43,44,42);(13,17,18,26)(14,20,15,22)(16,23,24,25)(19,27,21,28)(29,33,34,41)(30,36,32,40)(31,37,38,44)(35,42,39,43)Polycyclic presentation:
2^9|a^2=fh,b^2=gh,c^2=g,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2668618);