Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499823 |
| 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×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1788 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,3)(13,14,18,23)(15,19,26,33)(16,21,27,36)(17,20,29,22)(24,31,37,43)(25,30,39,32)(28,34,40,35)(38,41,44,42);(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,15,18,26)(14,19,23,33)(16,24,27,37)(17,25,29,39)(20,30,22,32)(21,31,36,43)(28,38,40,44)(34,41,35,42);(1,3,4,2)(6,9)(7,11)(14,20)(16,27)(17,29)(19,30)(21,35)(22,23)(24,37)(25,39)(31,42)(32,33)(34,36)(41,43);(5,7,8,11)(6,10,9,12)(13,16)(14,21)(15,24)(17,28)(18,27)(19,31)(20,34)(22,35)(23,36)(25,38)(26,37)(29,40)(30,41)(32,42)(33,43)(39,44);(6,9)(10,12)(16,27)(21,36)(24,37)(28,40)(31,43)(34,35)(38,44)(41,42)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=f,d^2=h,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,d^c=dhi,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6421680);