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 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2198 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,4,3)(7,11)(10,12)(13,14)(15,20)(16,22)(17,24)(18,23)(19,25)(21,30)(26,42)(27,35)(28,41)(29,37)(31,40)(32,36)(33,39)(34,44)(38,43);(2,3)(6,9)(7,11)(13,15,19,29)(14,20,25,37)(16,27,31,26)(17,32,18,28)(21,38,39,34)(22,35,40,42)(23,41,24,36)(30,43,33,44);(5,6)(7,10)(8,9)(11,12)(13,16,19,31)(14,21,25,39)(15,26,29,27)(17,30,18,33)(20,34,37,38)(22,24,40,23)(28,43,32,44)(35,41,42,36);(1,3,4,2)(6,9)(10,12)(13,16)(14,22)(15,27)(17,33)(18,30)(19,31)(20,35)(21,23)(24,39)(25,40)(26,29)(28,43)(32,44)(34,36)(37,42)(38,41);(5,7,8,11)(6,10,9,12)(13,17,19,18)(14,23,25,24)(15,28,29,32)(16,30,31,33)(20,36,37,41)(21,22,39,40)(26,43,27,44)(34,35,38,42)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,d^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7053111);