Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573591 |
| 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 | #33166 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1937 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)(13,14,19,33,18,23,17,22)(15,24,27,20,26,36,28,35)(16,29,30,21,32,37,31,38)(25,34,42,39,40,43,41,44);(1,2,3,4)(6,10)(8,11)(13,15)(14,20)(16,25)(17,27)(18,26)(19,28)(21,34)(22,36)(23,35)(24,33)(29,39)(30,41)(31,42)(32,40)(37,44)(38,43);(5,6)(7,8)(9,10)(11,12)(13,16,18,32)(14,21,23,38)(15,25,26,40)(17,30,19,31)(20,34,35,43)(22,37,33,29)(24,39,36,44)(27,41,28,42);(5,7,9,12)(6,11,10,8)(15,26)(20,35)(24,36)(25,40)(27,28)(34,43)(39,44)(41,42);(5,8,9,11)(6,7,10,12)(13,17,18,19)(14,22,23,33)(15,27,26,28)(16,30,32,31)(20,36,35,24)(21,37,38,29)(25,41,40,42)(34,44,43,39)Polycyclic presentation:
2^9|a^2=gi,b^2=fgi,c^2=i,d^2=h,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,g^c=gi,e^d=eh83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6528605);