Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499792 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3172 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)(15,25)(16,30)(17,24)(18,23)(19,22)(20,28)(21,34)(26,38)(27,36)(29,37)(31,35)(32,40)(33,39)(41,43)(42,44);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15,19,27)(14,20,22,37)(16,31,32,26)(17,29,18,28)(21,38,39,35)(23,36,24,25)(30,41,40,44)(33,42,34,43);(6,9)(7,11)(13,16)(14,21)(15,26)(17,34)(18,33)(19,32)(20,35)(22,39)(23,30)(24,40)(25,41)(27,31)(28,43)(29,42)(36,44)(37,38);(5,7,8,11)(6,10,9,12)(14,22)(15,27)(16,32)(23,24)(28,29)(33,34)(35,38)(41,44);(6,9)(10,12)(14,22)(15,27)(21,39)(23,24)(26,31)(28,29)(30,40)(42,43)Polycyclic presentation:
2^9|b^2=i,d^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6420032);