Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500701 |
| 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)(15,20)(16,29)(17,23)(18,22)(19,24)(21,30)(25,39)(26,35)(27,34)(28,38)(31,36)(32,37)(33,41)(40,43)(42,44);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,15)(14,20)(16,25)(17,27)(18,28)(19,26)(21,33)(22,38)(23,34)(24,35)(29,39)(30,41)(31,40)(32,42)(36,43)(37,44);(2,4)(13,16,19,31)(14,21,24,37)(15,25,26,40)(17,30,18,32)(20,33,35,44)(22,29,23,36)(27,41,28,42)(34,43,38,39);(6,11)(7,12)(14,22)(15,26)(17,18)(20,34)(21,29)(23,24)(25,40)(30,32)(33,43)(35,38)(36,37)(39,44);(5,8,9,10)(6,7,11,12)(13,17,19,18)(14,23,24,22)(15,27,26,28)(16,30,31,32)(20,34,35,38)(21,36,37,29)(25,41,40,42)(33,43,44,39)Polycyclic presentation:
2^9|c^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dfg,g^a=gi,c^b=ch,d^b=dfi,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, 7410056);