Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2760365 |
| 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 | |
1642 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)(14,18)(15,21)(16,23)(26,28)(29,30,33,34)(31,37,39,35)(32,36,41,42)(38,44,40,43);(5,6,8,10)(7,11,12,9)(14,18)(15,21)(20,25)(22,27)(30,34)(35,37)(36,42)(43,44);(5,7,8,12)(6,9,10,11)(13,14)(15,19)(16,20)(17,18)(21,24)(22,26)(23,25)(27,28)(29,30)(31,37)(32,36)(33,34)(35,39)(38,44)(40,43)(41,42);(1,2)(3,4)(13,15,17,21)(14,19,18,24)(16,22,23,27)(20,26,25,28)(29,31,33,39)(30,35,34,37)(32,40,41,38)(36,44,42,43);(13,16)(14,20)(15,22)(17,23)(18,25)(19,26)(21,27)(24,28)(29,32,33,41)(30,36,34,42)(31,38,39,40)(35,43,37,44)Polycyclic presentation:
2^9|a^2=f,b^2=gh,c^2=gh,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2669446);