Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1808 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)(7,11)(8,9)(10,12)(13,14,18,19)(15,16,21,20)(17,22,25,27)(23,24,28,26)(29,30)(31,36)(32,38)(33,35)(34,37)(39,43)(40,44)(41,42);(5,6)(7,11)(8,9)(10,12)(14,19)(15,21)(17,25)(24,26)(29,31,35,32)(30,36,33,38)(34,40,42,39)(37,44,41,43);(6,9)(10,11)(13,15,18,21)(14,20,19,16)(17,23,25,28)(22,26,27,24)(29,32,35,31)(30,36,33,38)(34,40,42,39)(37,43,41,44);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,16,18,20)(14,21,19,15)(17,26,25,24)(22,23,27,28)(29,33,35,30)(31,38,32,36)(34,41,42,37)(39,44,40,43);(5,7)(6,10)(8,12)(9,11)(13,17)(14,22)(15,23)(16,24)(18,25)(19,27)(20,26)(21,28)(29,34,35,42)(30,37,33,41)(31,39,32,40)(36,43,38,44)Polycyclic presentation:
2^9|a^2=f,b^2=gh,c^2=fgh,d^2=fgh,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5544624);