Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1911 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)(5,6,8,10)(7,9,11,12)(13,14,18,19)(15,16,20,24)(17,21,26,27)(22,23,28,25)(30,34)(31,37)(36,39)(38,40);(7,11)(9,12)(14,19)(15,20)(21,27)(22,28)(29,30,33,34)(31,35,37,42)(32,39,41,36)(38,44,40,43);(5,7,8,11)(6,9,10,12)(13,15)(14,16)(17,22)(18,20)(19,24)(21,23)(25,27)(26,28)(29,30)(31,35)(32,36)(33,34)(37,42)(38,43)(39,41)(40,44);(1,2)(3,4)(13,16)(14,20)(15,19)(17,25)(18,24)(21,22)(23,26)(27,28)(29,31,33,37)(30,35,34,42)(32,40,41,38)(36,44,39,43);(5,6)(7,9)(8,10)(11,12)(13,17)(14,21)(15,22)(16,23)(18,26)(19,27)(20,28)(24,25)(29,32,33,41)(30,36,34,39)(31,38,37,40)(35,43,42,44)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=fh,d^2=g,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5518919);