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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1868 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)(7,11)(10,12)(13,14,18,19)(15,20,21,16)(17,25,27,22)(23,24,26,28)(29,30,34,33)(31,37,39,35)(32,40,42,36)(38,43,41,44);(6,9)(10,12)(14,19)(16,20)(22,25)(24,28)(29,31,34,39)(30,35,33,37)(32,41,42,38)(36,44,40,43);(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,19)(17,26)(18,21)(22,24)(23,27)(25,28)(29,31)(30,35)(32,38)(33,37)(34,39)(36,43)(40,44)(41,42);(1,2,3,4)(13,16,18,20)(14,21,19,15)(17,24,27,28)(22,26,25,23)(29,32)(30,36)(31,38)(33,40)(34,42)(35,43)(37,44)(39,41);(5,7,8,11)(6,10,9,12)(13,17,18,27)(14,22,19,25)(15,23,21,26)(16,24,20,28)(29,33,34,30)(31,35,39,37)(32,40,42,36)(38,43,41,44)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=gi,e^2=fgh,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, 5189292);