Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533123 |
| 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 | C2×#546 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2023 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,3)(13,14)(15,20)(16,29)(17,22)(18,23)(19,24)(21,30)(25,39)(26,34)(27,35)(28,36)(31,37)(32,38)(33,41)(40,43)(42,44);(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,15,19,28)(14,20,24,36)(16,25,31,42)(17,27,18,26)(21,33,38,43)(22,35,23,34)(29,39,37,44)(30,41,32,40);(1,3,4,2)(6,9)(10,12)(13,16,19,31)(14,21,24,38)(15,25,28,42)(17,30,18,32)(20,33,36,43)(22,37,23,29)(26,40,27,41)(34,39,35,44);(5,7)(6,10)(8,11)(9,12)(13,17,19,18)(14,22,24,23)(15,26,28,27)(16,30,31,32)(20,34,36,35)(21,37,38,29)(25,40,42,41)(33,39,43,44);(7,11)(10,12)(13,17,19,18)(14,23,24,22)(15,27,28,26)(16,30,31,32)(20,34,36,35)(21,29,38,37)(25,41,42,40)(33,39,43,44)Polycyclic presentation:
2^9|b^2=fi,c^2=fi,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=di,g^c=gi,e^d=eh71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6447181);