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) = C26×C4 |
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 | |
1730 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)(15,16)(17,21)(18,19)(20,23)(22,24)(25,26)(27,28)(29,30,34,35)(31,38,36,32)(33,41,44,37)(39,40,42,43);(5,6,8,10)(7,9,11,12)(14,19)(16,23)(21,28)(24,26)(30,35)(31,36)(37,41)(39,42);(5,6)(7,9)(8,10)(11,12)(13,15,18,20)(14,16,19,23)(17,25,27,22)(21,26,28,24)(29,31,34,36)(30,32,35,38)(33,42,44,39)(37,43,41,40);(1,2,3,4)(5,7)(6,9)(8,11)(10,12)(13,16)(14,20)(15,19)(17,26)(18,23)(21,22)(24,27)(25,28)(29,32)(30,36)(31,35)(33,43)(34,38)(37,39)(40,44)(41,42);(13,17,18,27)(14,21,19,28)(15,22,20,25)(16,24,23,26)(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=fghi,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, 3377125);