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 | |
1718 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)(7,11)(9,12)(13,14)(15,19)(16,20)(17,18)(21,27)(22,26)(23,28)(24,25)(29,30,34,35)(31,38,36,32)(33,41,44,37)(39,40,42,43);(5,6,8,10)(7,9,11,12)(14,18)(19,26)(20,24)(27,28)(30,35)(31,36)(37,41)(39,42);(5,6)(7,9)(8,10)(11,12)(13,15,17,22)(14,19,18,26)(16,23,25,21)(20,28,24,27)(29,31,34,36)(30,32,35,38)(33,42,44,39)(37,43,41,40);(1,2)(3,4)(5,7,8,11)(6,9,10,12)(13,14)(15,19)(16,24)(17,18)(20,25)(21,28)(22,26)(23,27)(29,32)(30,36)(31,35)(33,43)(34,38)(37,39)(40,44)(41,42);(13,16,17,25)(14,20,18,24)(15,21,22,23)(19,27,26,28)(29,33)(30,37)(31,39)(32,40)(34,44)(35,41)(36,42)(38,43)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=f,d^2=gh,e^2=fg,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, 3251037);