Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #23854 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2156 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,10)(8,9)(11,12)(13,14,19,24)(15,20,27,35)(16,22,29,38)(17,23,18,21)(25,33,41,44)(26,32,30,34)(28,36,31,37)(39,43,40,42);(1,2)(3,4)(6,9)(10,12)(13,15)(14,20)(16,25)(17,30)(18,26)(19,27)(21,32)(22,33)(23,34)(24,35)(28,40)(29,41)(31,39)(36,42)(37,43)(38,44);(2,4)(14,21)(17,18)(20,32)(22,36)(23,24)(26,30)(28,31)(33,42)(34,35)(37,38)(39,40)(43,44);(2,4)(13,16,19,29)(14,22,24,38)(15,25,27,41)(17,31,18,28)(20,33,35,44)(21,36,23,37)(26,40,30,39)(32,42,34,43);(5,7,8,11)(6,10,9,12)(13,17,19,18)(14,23,24,21)(15,26,27,30)(16,28,29,31)(20,32,35,34)(22,36,38,37)(25,39,41,40)(33,43,44,42)Polycyclic presentation:
2^9|a^2=i,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,e^c=ei,g^c=gi,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6347223);