Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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 | |
1671 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)(15,20,22,16)(17,21,26,27)(23,28,25,24)(30,34)(31,37)(36,40)(39,43);(6,9)(10,11)(14,19)(15,22)(21,27)(23,25)(29,30,33,34)(31,38,37,35)(32,36,41,40)(39,44,43,42);(5,7,8,12)(6,10,9,11)(13,15)(14,20)(16,19)(17,25)(18,22)(21,24)(23,26)(27,28)(29,30)(31,35)(32,40)(33,34)(36,41)(37,38)(39,44)(42,43);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,16,18,20)(14,15,19,22)(17,24,26,28)(21,23,27,25)(29,31,33,37)(30,35,34,38)(32,39,41,43)(36,42,40,44);(5,7)(6,10)(8,12)(9,11)(13,17,18,26)(14,21,19,27)(15,23,22,25)(16,24,20,28)(29,32)(30,36)(31,39)(33,41)(34,40)(35,42)(37,43)(38,44)Polycyclic presentation:
2^9|a^2=fg,b^2=fgh,c^2=fh,d^2=h,e^2=fg,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, 4509267);