Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 | C2×#449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1823 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)(13,14,18,24,19,23,33,34)(15,25,27,36,16,21,29,20)(17,22,32,39,31,40,41,44)(26,38,30,43,28,37,42,35);(6,9)(7,11)(13,15)(14,20)(16,19)(17,30)(18,29)(21,24)(22,38)(23,36)(25,34)(26,32)(27,33)(28,41)(31,42)(35,39)(37,40)(43,44);(2,4)(5,6)(7,10)(8,9)(11,12)(13,16,19,15)(14,21,23,25)(17,28,31,26)(18,29,33,27)(20,34,36,24)(22,37,40,38)(30,32,42,41)(35,44,43,39);(6,9)(10,12)(13,17,18,32,19,31,33,41)(14,22,24,39,23,40,34,44)(15,26,27,30,16,28,29,42)(20,35,25,38,36,43,21,37);(5,7,8,11)(6,10,9,12)(14,23)(15,16)(17,31)(24,34)(27,29)(32,41)(35,43)(37,38)Polycyclic presentation:
2^9|a^2=fg,c^2=i,d^2=f,e^2=hi,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=dhi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7058632);