Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | |
1714 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)(7,11)(9,12)(13,14,18,19)(15,20,23,27)(16,24,25,17)(21,22,28,26)(30,34)(31,37)(32,39)(43,44);(5,6,8,10)(7,9,11,12)(14,19)(16,25)(20,27)(21,28)(29,30)(31,35)(32,36)(33,34)(37,42)(38,43)(39,41)(40,44);(5,6)(7,9)(8,10)(11,12)(13,15)(14,20)(16,21)(17,26)(18,23)(19,27)(22,24)(25,28)(29,31,33,37)(30,35,34,42)(32,40,39,38)(36,44,41,43);(1,2,3,4)(5,7)(6,9)(8,11)(10,12)(13,16,18,25)(14,17,19,24)(15,21,23,28)(20,26,27,22)(29,30)(31,35)(32,41)(33,34)(36,39)(37,42)(38,44)(40,43);(13,17)(14,16)(15,22)(18,24)(19,25)(20,28)(21,27)(23,26)(29,32,33,39)(30,36,34,41)(31,38,37,40)(35,43,42,44)Polycyclic presentation:
2^9|a^2=fg,b^2=h,c^2=g,d^2=ghi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3377162);