Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438943 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2112 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(5,6,8,10)(7,9,11,12)(13,14)(15,20)(16,28)(17,22)(18,32)(19,23)(21,30)(24,37)(25,34)(26,35)(27,41)(29,36)(31,42)(33,40)(38,43)(39,44);(2,4)(5,7)(6,9)(8,11)(10,12)(13,15)(14,20)(16,24)(17,25)(18,27)(19,26)(21,33)(22,34)(23,35)(28,37)(29,38)(30,40)(31,39)(32,41)(36,43)(42,44);(7,11)(9,12)(13,16,18,30,19,31,17,29)(14,21,22,28,23,36,32,42)(15,24,27,40,26,39,25,38)(20,33,34,37,35,43,41,44);(7,11)(9,12)(13,17,19,18)(14,22,23,32)(15,25,26,27)(16,29,31,30)(20,34,35,41)(21,28,36,42)(24,38,39,40)(33,37,43,44);(15,26)(20,35)(24,39)(25,27)(33,43)(34,41)(37,44)(38,40)Polycyclic presentation:
2^9|a^2=h,c^2=g,d^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6364358);