Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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 | |
1827 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)(7,9)(8,10)(11,12)(13,14)(15,16)(17,25)(18,19)(20,23)(21,27)(22,28)(24,26)(30,34)(35,37)(36,42)(43,44);(8,12)(10,11)(14,19)(16,23)(21,25)(24,28)(29,30,33,34)(31,37,39,35)(32,36,41,42)(38,44,40,43);(6,7)(10,11)(13,15)(14,16)(17,26)(18,20)(19,23)(21,28)(22,27)(24,25)(29,31,33,39)(30,35,34,37)(32,40,41,38)(36,44,42,43);(1,2,3,4)(5,7,9,6)(8,11,12,10)(13,16)(14,20)(15,19)(17,24)(18,23)(21,26)(22,25)(27,28)(29,30)(31,37)(32,36)(33,34)(35,39)(38,44)(40,43)(41,42);(5,8,9,12)(6,10,7,11)(13,17,18,27)(14,21,19,25)(15,22,20,26)(16,24,23,28)(29,32,33,41)(30,36,34,42)(31,38,39,40)(35,43,37,44)Polycyclic presentation:
2^9|b^2=h,c^2=h,d^2=fi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5187955);