Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573591 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3521 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2998 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)(13,14,17,22,19,24,18,23)(15,25,27,35,28,36,29,20)(16,30,32,21,33,37,31,38)(26,40,43,39,42,34,41,44);(2,4)(5,6,8,9)(7,10,11,12)(13,15)(14,20)(16,26)(17,29)(18,27)(19,28)(21,34)(22,36)(23,25)(24,35)(30,44)(31,43)(32,41)(33,42)(37,39)(38,40);(6,9)(7,11)(13,16)(14,21)(15,26)(17,31)(18,32)(19,33)(20,34)(22,30)(23,37)(24,38)(25,39)(27,41)(28,42)(29,43)(35,40)(36,44);(5,7)(6,10)(8,11)(9,12)(13,17,19,18)(14,22,24,23)(15,27,28,29)(16,31,33,32)(20,25,35,36)(21,30,38,37)(26,41,42,43)(34,39,40,44);(6,9)(7,11)(15,28)(20,35)(25,36)(26,42)(27,29)(34,40)(39,44)(41,43)Polycyclic presentation:
2^9|a^2=f,b^2=h,d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=ehi,f^b=fi,g^b=gi,d^c=dh,f^c=fi,g^c=gi,e^d=eh83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6503476);