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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2333 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,4,3)(5,6)(7,9)(8,10)(11,12)(13,14)(15,20)(16,22)(17,23)(18,32)(19,24)(21,29)(25,34)(26,35)(27,42)(28,36)(30,37)(31,38)(33,39)(40,43)(41,44);(2,3)(5,6)(7,11)(8,10)(9,12)(13,15,19,28)(14,20,24,36)(16,25,30,40)(17,26,18,27)(21,33,38,44)(22,34,37,43)(23,35,32,42)(29,39,31,41);(5,7)(6,9)(8,12)(10,11)(13,16,19,30)(14,21,24,38)(15,25,28,40)(17,29,18,31)(20,33,36,44)(22,23,37,32)(26,39,27,41)(34,35,43,42);(1,3,4,2)(7,12)(9,11)(13,16)(14,22)(15,25)(17,31)(18,29)(19,30)(20,34)(21,32)(23,38)(24,37)(26,41)(27,39)(28,40)(33,42)(35,44)(36,43);(6,10)(9,11)(13,17,19,18)(14,23,24,32)(15,26,28,27)(16,29,30,31)(20,35,36,42)(21,37,38,22)(25,39,40,41)(33,43,44,34)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6903720);