Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432843 |
| 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×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2105 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)(5,6)(7,9)(8,10)(11,12)(13,14,19,23)(15,20,27,36)(16,28,30,37)(17,22,18,31)(21,32,38,29)(24,39,42,43)(25,34,26,35)(33,41,44,40);(1,2,4,3)(5,7)(6,9)(8,11)(10,12)(13,15)(14,20)(16,24)(17,25)(18,26)(19,27)(21,33)(22,34)(23,36)(28,39)(29,40)(30,42)(31,35)(32,41)(37,43)(38,44);(1,3,4,2)(7,11)(9,12)(13,16)(14,21)(15,24)(17,29)(18,32)(19,30)(20,33)(22,37)(23,38)(25,40)(26,41)(27,42)(28,31)(34,43)(35,39)(36,44);(6,10)(9,12)(13,17,19,18)(14,22,23,31)(15,25,27,26)(16,29,30,32)(20,34,36,35)(21,37,38,28)(24,40,42,41)(33,43,44,39);(13,17,19,18)(14,22,23,31)(15,26,27,25)(16,29,30,32)(20,35,36,34)(21,37,38,28)(24,41,42,40)(33,39,44,43)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=f,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,c^b=ch,e^b=ei,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6340410);