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 | |
1614 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)(5,6,9,8)(7,10,11,12)(13,14)(15,19)(16,18)(17,20)(21,23)(22,27)(24,26)(25,28)(30,34)(31,37)(32,39)(43,44);(7,11)(10,12)(14,16)(19,21)(20,24)(27,28)(29,30,33,34)(31,35,37,42)(32,36,39,40)(38,43,41,44);(5,7,9,11)(6,10,8,12)(13,15,18,23)(14,19,16,21)(17,25,26,22)(20,28,24,27)(29,30)(31,35)(32,40)(33,34)(36,39)(37,42)(38,44)(41,43);(1,2,3,4)(5,6,9,8)(7,10,11,12)(13,16,18,14)(15,21,23,19)(17,20,26,24)(22,27,25,28)(29,31)(30,35)(32,41)(33,37)(34,42)(36,44)(38,39)(40,43);(5,8,9,6)(7,12,11,10)(13,17)(14,20)(15,22)(16,24)(18,26)(19,27)(21,28)(23,25)(29,32)(30,36)(31,38)(33,39)(34,40)(35,43)(37,41)(42,44)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=fh,d^2=hi,e^2=gh,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, 3387085);