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 | #30414 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1863 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)(15,24)(16,28)(17,22)(18,23)(19,33)(20,27)(21,31)(25,40)(26,35)(29,38)(30,37)(32,36)(34,43)(39,41)(42,44);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,15)(14,20)(16,25)(17,32)(18,27)(19,26)(21,34)(22,35)(23,24)(28,44)(29,43)(30,42)(31,41)(33,36)(37,40)(38,39);(2,4)(13,16,18,30)(14,21,23,38)(15,25,27,42)(17,31,19,29)(20,34,24,39)(22,37,33,28)(26,43,32,41)(35,40,36,44);(6,11)(7,12)(14,22)(17,19)(20,35)(21,37)(23,33)(24,36)(26,32)(28,38)(29,31)(34,40)(39,44)(41,43);(5,8,9,10)(6,7,11,12)(13,17,18,19)(14,22,23,33)(15,26,27,32)(16,29,30,31)(20,36,24,35)(21,28,38,37)(25,41,42,43)(34,40,39,44)Polycyclic presentation:
2^9|c^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dfg,g^a=gi,c^b=ch,d^b=dfi,e^b=ei,e^c=ei,g^c=gi,e^d=ei,g^d=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7410247);