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)(22,35)(23,32)(24,37)(26,31)(28,38)(29,33)(36,40)(39,44)(41,42);(5,6)(7,11)(8,10)(9,12)(13,14,19,23)(15,24,27,21)(16,28,30,35)(17,20,18,32)(22,33,38,29)(25,40,43,44)(26,34,31,37)(36,42,39,41);(2,4)(5,7,8,12)(6,9,10,11)(13,15,19,27)(14,21,23,24)(16,25,30,43)(17,31,18,26)(20,34,32,37)(22,36,38,39)(28,44,35,40)(29,42,33,41);(7,12)(9,11)(13,16)(14,22)(15,25)(17,29)(18,33)(19,30)(20,35)(21,36)(23,38)(24,39)(26,41)(27,43)(28,32)(31,42)(34,40)(37,44);(6,10)(9,11)(13,17,19,18)(14,20,23,32)(15,26,27,31)(16,29,30,33)(21,37,24,34)(22,35,38,28)(25,41,43,42)(36,44,39,40)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=df,e^b=ehi,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, 6970004);