Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500689 |
| 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 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2160 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6,9,12)(7,8,11,10)(13,14,19,21,18,24,17,23)(15,20,26,33,28,36,27,35)(16,29,30,22,32,37,31,38)(25,39,42,34,41,44,40,43);(1,2)(3,4)(5,7,9,11)(6,10,12,8)(13,15,18,28)(14,20,24,36)(16,25,32,41)(17,27,19,26)(21,33,23,35)(22,34,38,43)(29,39,37,44)(30,42,31,40);(2,4)(5,8,9,10)(6,11,12,7)(14,21)(17,19)(20,33)(22,29)(23,24)(26,27)(30,31)(34,39)(35,36)(37,38)(40,42)(43,44);(6,12)(7,11)(13,16)(14,22)(15,25)(17,30)(18,32)(19,31)(20,34)(21,29)(23,37)(24,38)(26,40)(27,42)(28,41)(33,39)(35,44)(36,43);(13,17,18,19)(14,23,24,21)(15,26,28,27)(16,30,32,31)(20,33,36,35)(22,37,38,29)(25,40,41,42)(34,39,43,44)Polycyclic presentation:
2^9|a^2=fgi,b^2=fi,c^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,e^b=ei,e^c=ei,g^c=gi,g^d=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7411014);