Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
1770 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)(5,6,8,10)(7,9,11,12)(13,14,18,16)(15,21,23,19)(17,24,26,20)(22,27,25,28)(29,30)(31,32)(33,37)(34,35)(36,40)(38,39)(41,44)(42,43);(5,7,8,11)(6,9,10,12)(14,16)(19,21)(20,24)(27,28)(30,35)(31,36)(37,43)(38,41);(5,7)(6,9)(8,11)(10,12)(13,15,18,23)(14,19,16,21)(17,25,26,22)(20,28,24,27)(29,31,34,36)(30,32,35,40)(33,41,42,38)(37,44,43,39);(1,2)(3,4)(13,16,18,14)(15,21,23,19)(17,24,26,20)(22,28,25,27)(29,32)(30,36)(31,35)(33,39)(34,40)(37,41)(38,43)(42,44);(5,6)(7,9)(8,10)(11,12)(13,17,18,26)(14,20,16,24)(15,22,23,25)(19,27,21,28)(29,33,34,42)(30,37,35,43)(31,38,36,41)(32,39,40,44)Polycyclic presentation:
2^9|a^2=fh,b^2=fgh,c^2=f,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4506021);