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 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2134 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6)(7,9)(8,10)(11,12)(13,14,19,24)(15,25,26,38)(16,21,30,36)(17,23,18,22)(20,32,33,28)(27,39,40,44)(29,37,31,35)(34,41,43,42);(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(15,26)(20,33)(25,38)(27,40)(28,32)(34,43)(39,44)(41,42);(5,7,8,12)(6,9,10,11)(13,15,19,26)(14,20,24,33)(16,27,30,40)(17,28,18,32)(21,34,36,43)(22,25,23,38)(29,41,31,42)(35,44,37,39);(2,4)(7,12)(9,11)(13,16)(14,21)(15,27)(17,31)(18,29)(19,30)(20,34)(22,37)(23,35)(24,36)(25,39)(26,40)(28,42)(32,41)(33,43)(38,44);(6,10)(9,11)(13,17,19,18)(14,22,24,23)(15,28,26,32)(16,29,30,31)(20,25,33,38)(21,35,36,37)(27,41,40,42)(34,44,43,39)Polycyclic presentation:
2^9|a^2=i,c^2=h,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6961512);