Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996355 |
| 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 | #24129 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1832 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)(5,6)(7,10)(8,11)(9,12)(13,14,18,22,19,23,31,37)(15,24,26,33,27,36,17,20)(16,21,30,35,29,34,41,43)(25,38,40,42,39,44,28,32);(5,6)(7,10)(8,11)(9,12)(13,15,19,27)(14,20,23,33)(16,25,29,39)(17,31,26,18)(21,32,34,42)(22,36,37,24)(28,41,40,30)(35,44,43,38);(1,2,3,4)(5,7,10,6)(8,11,9,12)(13,16)(14,21)(15,25)(17,28)(18,30)(19,29)(20,32)(22,35)(23,34)(24,38)(26,40)(27,39)(31,41)(33,42)(36,44)(37,43);(5,8,10,9)(6,11,7,12)(13,17,18,15,19,26,31,27)(14,20,22,24,23,33,37,36)(16,28,30,25,29,40,41,39)(21,32,35,38,34,42,43,44);(5,9,10,8)(6,11,7,12)(16,29)(21,34)(25,39)(28,40)(30,41)(32,42)(35,43)(38,44)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=g,d^2=fh,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7106132);