Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| 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, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1943 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)(7,11)(10,12)(13,14)(15,16)(17,25)(18,19)(20,24)(21,27)(22,26)(23,28)(30,31)(32,39)(36,38)(40,41);(6,9)(10,12)(14,19)(15,20)(21,25)(22,28)(29,30,34,31)(32,37,39,35)(33,36,42,38)(40,44,41,43);(5,6)(7,10)(8,9)(11,12)(13,15,18,20)(14,16,19,24)(17,22,27,28)(21,23,25,26)(29,31,34,30)(32,37,39,35)(33,38,42,36)(40,44,41,43);(1,2)(3,4)(13,16)(14,20)(15,19)(17,26)(18,24)(21,22)(23,27)(25,28)(29,32,34,39)(30,35,31,37)(33,41,42,40)(36,44,38,43);(5,7)(6,10)(8,11)(9,12)(13,17,18,27)(14,21,19,25)(15,22,20,28)(16,23,24,26)(29,33,34,42)(30,36,31,38)(32,40,39,41)(35,43,37,44)Polycyclic presentation:
2^9|b^2=g,c^2=f,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4768712);