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×#3524 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2974 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)(13,14,17,33,19,23,18,22)(15,24,26,20,27,40,28,36)(16,29,34,21,32,38,31,39)(25,35,43,41,30,37,42,44);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,30)(17,28)(18,26)(19,27)(21,37)(22,40)(23,36)(24,33)(25,32)(29,44)(31,42)(34,43)(35,39)(38,41);(6,9)(10,12)(13,16)(14,21)(15,25)(17,31)(18,34)(19,32)(20,35)(22,38)(23,39)(24,41)(26,42)(27,30)(28,43)(29,33)(36,37)(40,44);(6,9)(10,12)(13,17,19,18)(14,22,23,33)(15,26,27,28)(16,31,32,34)(20,24,36,40)(21,38,39,29)(25,42,30,43)(35,41,37,44);(5,7,8,11)(6,10,9,12)(14,23)(15,27)(21,39)(22,33)(25,30)(26,28)(29,38)(42,43)Polycyclic presentation:
2^9|a^2=fi,d^2=i,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,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, 6398567);