Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432843 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2942 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4)(7,11)(10,12)(13,14)(15,25)(16,30)(17,23)(18,24)(19,22)(20,28)(21,33)(26,38)(27,36)(29,37)(31,35)(32,39)(34,40)(41,43)(42,44);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15,19,29)(14,20,22,36)(16,31,34,26)(17,28,18,27)(21,38,39,35)(23,37,24,25)(30,41,40,44)(32,42,33,43);(6,9)(10,12)(13,16)(14,21)(15,26)(17,32)(18,33)(19,34)(20,35)(22,39)(23,30)(24,40)(25,41)(27,42)(28,43)(29,31)(36,38)(37,44);(5,7,8,11)(6,10,9,12)(14,22)(20,36)(21,39)(23,24)(25,37)(30,40)(35,38)(41,44);(13,17,19,18)(14,23,22,24)(15,27,29,28)(16,32,34,33)(20,25,36,37)(21,30,39,40)(26,42,31,43)(35,41,38,44)Polycyclic presentation:
2^9|a^2=fg,b^2=i,d^2=hi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,g^a=gi,c^b=ch,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6342650);