Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667932 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2315 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)(15,19)(16,21)(17,20)(18,22)(23,30)(24,32)(25,31)(26,33)(27,36)(28,38)(29,35)(34,40)(37,42)(39,43)(41,44);(2,3)(6,9)(7,11)(13,15,18,26)(14,19,22,33)(16,27,28,24)(17,25,30,40)(20,31,23,34)(21,36,38,32)(29,41,42,39)(35,44,37,43);(1,2,4,3)(14,20)(17,30)(19,31)(21,35)(22,23)(25,40)(29,42)(32,43)(33,34)(36,44)(37,38)(39,41);(6,9)(10,12)(13,16,18,28)(14,21,22,38)(15,24,26,27)(17,29,30,42)(19,32,33,36)(20,35,23,37)(25,39,40,41)(31,43,34,44);(1,3,4,2)(5,7,8,11)(6,10,9,12)(14,22)(16,28)(19,33)(20,23)(24,27)(29,42)(31,34)(39,41)Polycyclic presentation:
2^9|b^2=i,c^2=h,d^2=i,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6716661);