Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4417862 |
| 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 | |
1682 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)(5,6,10,9)(7,12,11,8)(13,14,18,19)(15,16,21,24)(17,25,27,20)(22,26,28,23)(30,34)(31,37)(36,39)(38,43);(5,6,10,9)(7,8,11,12)(14,19)(15,21)(20,25)(22,28)(29,30)(31,35)(32,36)(33,34)(37,41)(38,42)(39,40)(43,44);(5,7)(6,8)(9,12)(10,11)(13,15)(14,16)(17,22)(18,21)(19,24)(20,23)(25,26)(27,28)(29,31,33,37)(30,35,34,41)(32,38,40,43)(36,42,39,44);(1,2,3,4)(5,8,10,12)(6,11,9,7)(13,16)(14,15)(17,26)(18,24)(19,21)(20,28)(22,25)(23,27)(29,30)(31,35)(32,39)(33,34)(36,40)(37,41)(38,44)(42,43);(5,9,10,6)(7,12,11,8)(13,17,18,27)(14,20,19,25)(15,22,21,28)(16,23,24,26)(29,32)(30,36)(31,38)(33,40)(34,39)(35,42)(37,43)(41,44)Polycyclic presentation:
2^9|a^2=fg,b^2=fgh,c^2=fh,d^2=fi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4341941);