Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437770 |
| 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 | |
2027 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,10)(8,9)(11,12)(13,14,20,35,19,25,18,24)(15,26,34,21,30,40,29,38)(16,31,17,23,32,22,33,39)(27,37,43,44,42,41,28,36);(1,2)(3,4)(13,15)(14,21)(16,27)(17,28)(18,34)(19,30)(20,29)(22,36)(23,37)(24,40)(25,38)(26,35)(31,44)(32,42)(33,43)(39,41);(2,4)(6,9)(10,12)(13,16,19,32)(14,22,25,31)(15,27,30,42)(17,18,33,20)(21,36,38,44)(23,35,39,24)(26,41,40,37)(28,34,43,29);(2,4)(13,17)(14,23)(15,28)(16,18)(19,33)(20,32)(21,37)(22,35)(24,31)(25,39)(26,36)(27,34)(29,42)(30,43)(38,41)(40,44);(5,7,8,11)(6,10,9,12)(16,32)(17,33)(22,31)(23,39)(27,42)(28,43)(36,44)(37,41)Polycyclic presentation:
2^9|a^2=fi,c^2=i,e^2=gi,f^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,e^c=ei,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6361624);