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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2153 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)(15,21)(16,22)(17,23)(18,20)(19,24)(25,34)(26,32)(27,36)(28,33)(29,37)(30,38)(31,35)(39,42)(40,44)(41,43);(5,6)(7,11)(8,10)(9,12)(14,20)(16,28)(17,18)(21,32)(23,24)(25,39)(26,31)(29,30)(33,37)(35,36)(40,41)(42,44);(2,4)(5,7,8,12)(6,9,10,11)(13,15)(14,21)(16,25)(17,31)(18,26)(19,27)(20,32)(22,34)(23,35)(24,36)(28,39)(29,40)(30,41)(33,42)(37,44)(38,43);(7,12)(9,11)(13,16,19,30)(14,22,24,38)(15,25,27,41)(17,29,18,28)(20,33,23,37)(21,34,36,43)(26,39,31,40)(32,42,35,44);(6,10)(9,11)(13,17,19,18)(14,23,24,20)(15,26,27,31)(16,29,30,28)(21,32,36,35)(22,37,38,33)(25,39,41,40)(34,42,43,44)Polycyclic presentation:
2^9|c^2=h,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=dfi,e^b=ehi,f^b=fi,d^c=dh,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, 6961946);