Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| 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 | #45092 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1890 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)(5,6)(7,9)(8,10)(11,12)(13,14)(15,26)(16,22)(17,23)(18,24)(19,34)(20,25)(21,29)(27,41)(28,36)(30,37)(31,38)(32,40)(33,39)(35,42)(43,44);(5,6)(7,11)(8,10)(9,12)(13,15,18,28,20,17,19,29)(14,21,34,23,25,36,24,26)(16,27,33,43,32,30,31,42)(22,35,38,37,40,44,39,41);(2,4)(5,7)(6,9)(8,12)(10,11)(13,16,20,32)(14,22,25,40)(15,27,17,30)(18,33,19,31)(21,35,36,44)(23,37,26,41)(24,39,34,38)(28,43,29,42);(7,12)(9,11)(13,17,20,15)(14,23,25,26)(16,30,32,27)(18,28,19,29)(21,24,36,34)(22,37,40,41)(31,42,33,43)(35,39,44,38);(6,10)(9,11)(13,18,20,19)(14,24,25,34)(15,28,17,29)(16,31,32,33)(21,26,36,23)(22,38,40,39)(27,42,30,43)(35,37,44,41)Polycyclic presentation:
2^9|a^2=g,b^2=fi,c^2=i,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=ei,e^d=ei,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6910402);