Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1624 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(13,14,18,22)(15,16,20,19)(17,25,26,21)(23,27,28,24)(29,30)(31,37)(32,35)(33,36)(34,39)(38,44)(40,42)(41,43);(6,9)(10,12)(15,20)(16,19)(23,28)(24,27)(29,30,33,36)(31,37,39,34)(32,35,42,40)(38,44,41,43);(5,6)(7,10)(8,9)(11,12)(13,15,18,20)(14,19,22,16)(17,23,26,28)(21,27,25,24)(29,30)(31,34)(32,40)(33,36)(35,42)(37,39)(38,44)(41,43);(1,2)(3,4)(13,16,18,19)(14,20,22,15)(17,24,26,27)(21,28,25,23)(29,31,33,39)(30,34,36,37)(32,41,42,38)(35,44,40,43);(5,7,8,11)(6,10,9,12)(13,17,18,26)(14,21,22,25)(15,23,20,28)(16,24,19,27)(29,32,33,42)(30,35,36,40)(31,38,39,41)(34,43,37,44)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=g,d^2=fg,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3256723);