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 |
| prime radical | #45245 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2110 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)(15,25,26,40)(16,21,31,38)(17,23,18,22)(20,33,34,28)(27,41,29,44)(30,39,32,37)(35,42,36,43);(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(15,26)(20,34)(25,40)(27,29)(28,33)(35,36)(41,44)(42,43);(5,7,8,12)(6,9,10,11)(13,15)(14,20)(16,29)(17,28)(18,33)(19,26)(21,36)(22,25)(23,40)(24,34)(27,31)(30,43)(32,42)(35,38)(37,41)(39,44);(2,4)(7,12)(9,11)(13,16)(14,21)(15,27)(17,32)(18,30)(19,31)(20,35)(22,39)(23,37)(24,38)(25,41)(26,29)(28,43)(33,42)(34,36)(40,44);(6,10)(9,11)(13,17,19,18)(14,22,24,23)(15,28,26,33)(16,30,31,32)(20,25,34,40)(21,37,38,39)(27,42,29,43)(35,44,36,41)Polycyclic presentation:
2^9|a^2=i,c^2=hi,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=dh,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, 6961260);