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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1618 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)(5,6,8,10)(7,9,11,12)(13,14,17,18)(15,19,23,21)(16,24,26,20)(22,28,25,27)(29,30)(31,37)(32,40)(33,34)(35,38)(36,41)(39,42)(43,44);(5,7,8,11)(6,9,10,12)(14,18)(19,21)(20,24)(27,28)(30,34)(31,38)(32,41)(42,43);(5,7)(6,9)(8,11)(10,12)(13,15,17,23)(14,19,18,21)(16,25,26,22)(20,28,24,27)(29,31)(30,35)(32,39)(33,38)(34,37)(36,42)(40,43)(41,44);(1,2,3,4)(13,14)(15,21)(16,20)(17,18)(19,23)(22,28)(24,26)(25,27)(29,32)(30,36)(31,39)(33,41)(34,40)(35,42)(37,43)(38,44);(5,6)(7,9)(8,10)(11,12)(13,16,17,26)(14,20,18,24)(15,22,23,25)(19,27,21,28)(29,30)(31,37)(32,36)(33,34)(35,38)(39,43)(40,41)(42,44)Polycyclic presentation:
2^9|a^2=fh,b^2=gh,c^2=fg,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, 5186360);