Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470137 |
| 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×#121 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2926 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)(13,14,17,23,19,22,18,24)(15,25,28,36,27,37,29,20)(16,30,32,21,34,40,33,39)(26,38,42,41,31,35,43,44);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15,19,27)(14,20,22,36)(16,31,34,26)(17,29,18,28)(21,38,39,35)(23,37,24,25)(30,41,40,44)(32,42,33,43);(6,9)(10,12)(13,16,19,34)(14,21,22,39)(15,26,27,31)(17,33,18,32)(20,35,36,38)(23,30,24,40)(25,41,37,44)(28,43,29,42);(6,9)(10,12)(15,27)(20,36)(25,37)(26,31)(28,29)(35,38)(41,44)(42,43);(5,7,8,11)(6,10,9,12)(14,22)(15,27)(21,39)(23,24)(26,31)(28,29)(30,40)(42,43)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,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, 6398546);