Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| 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×#550 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1867 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)(14,20)(17,18)(21,34)(22,29)(23,24)(25,38)(27,32)(30,33)(35,44)(36,37)(39,40)(41,42);(5,6)(7,11)(8,10)(9,12)(13,14,19,24)(15,25,28,21)(16,29,31,36)(17,20,18,23)(22,30,37,33)(26,40,43,44)(27,34,32,38)(35,41,39,42);(2,4)(5,7,8,12)(6,9,10,11)(13,15,19,28)(14,21,24,25)(16,26,31,43)(17,32,18,27)(20,34,23,38)(22,35,37,39)(29,44,36,40)(30,42,33,41);(7,12)(9,11)(13,16)(14,22)(15,26)(17,30)(18,33)(19,31)(20,29)(21,35)(23,36)(24,37)(25,39)(27,41)(28,43)(32,42)(34,44)(38,40);(6,10)(9,11)(13,17,19,18)(14,23,24,20)(15,27,28,32)(16,30,31,33)(21,34,25,38)(22,36,37,29)(26,41,43,42)(35,44,39,40)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,f^b=fi,d^c=dhi,e^c=ei,f^d=fi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6970196);