Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| 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×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2192 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)(7,8)(9,11)(10,12)(13,14,17,22)(15,24,26,19)(16,28,30,37)(18,20,32,23)(21,31,39,33)(25,40,29,44)(27,36,42,34)(35,43,38,41);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,15)(14,19)(16,29)(17,26)(18,27)(20,34)(21,38)(22,24)(23,36)(25,30)(28,40)(31,43)(32,42)(33,41)(35,39)(37,44);(2,4)(5,8,9,10)(6,7,11,12)(14,20)(18,32)(19,34)(21,37)(22,23)(24,36)(27,42)(28,39)(31,33)(35,40)(38,44)(41,43);(6,11)(7,12)(13,16,17,30)(14,21,22,39)(15,25,26,29)(18,33,32,31)(19,35,24,38)(20,37,23,28)(27,43,42,41)(34,40,36,44);(15,26)(19,24)(25,29)(27,42)(34,36)(35,38)(40,44)(41,43)Polycyclic presentation:
2^9|a^2=i,c^2=fi,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,g^a=gi,c^b=ch,d^b=df,e^b=ei,g^c=gi,g^d=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7408480);