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 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1924 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)(15,23)(16,20)(17,32)(18,21)(19,27)(22,24)(25,41)(26,37)(28,42)(29,35)(30,39)(31,40)(33,38)(34,44)(36,43);(5,6)(7,11)(8,10)(9,12)(13,15,24,43,18,28,17,27)(14,19,32,42,21,36,22,23)(16,29,33,44,30,25,31,26)(20,37,40,41,39,34,38,35);(2,4)(5,7)(6,9)(8,12)(10,11)(13,16,18,30)(14,20,21,39)(15,25,28,29)(17,31,24,33)(19,34,36,37)(22,38,32,40)(23,41,42,35)(26,27,44,43);(7,12)(9,11)(13,16)(14,20)(15,26)(17,33)(18,30)(19,35)(21,39)(22,40)(23,37)(24,31)(25,43)(27,29)(28,44)(32,38)(34,42)(36,41);(6,10)(9,11)(14,21)(16,30)(19,36)(22,32)(23,42)(25,29)(26,44)(31,33)Polycyclic presentation:
2^9|a^2=g,b^2=fi,c^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,d^c=dh,e^c=ei,e^d=ei,f^d=fi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6910783);