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 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2116 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)(15,20)(16,21)(17,30)(18,22)(19,26)(23,33)(24,36)(25,37)(27,44)(28,41)(29,39)(31,35)(32,40)(34,43)(38,42);(5,6)(7,11)(8,10)(9,12)(13,15,18,25)(14,19,22,35)(16,27,29,43)(17,31,33,26)(20,23,37,30)(21,38,39,36)(24,32,42,28)(34,41,44,40);(2,4)(5,7,8,12)(6,9,10,11)(13,15)(14,20)(16,24)(17,26)(18,25)(19,30)(21,36)(22,37)(23,35)(27,28)(29,42)(31,33)(32,43)(34,40)(38,39)(41,44);(7,12)(9,11)(13,16,18,29)(14,21,22,39)(15,24,25,42)(17,32,33,28)(19,34,35,44)(20,36,37,38)(23,41,30,40)(26,43,31,27);(6,10)(9,11)(14,22)(15,25)(21,39)(23,30)(24,42)(26,31)(27,43)(40,41)Polycyclic presentation:
2^9|b^2=i,c^2=hi,d^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=eh,f^b=fi,d^c=dhi,e^c=ei,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6961765);