Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500120 |
| 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×#3561 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2924 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6)(7,10)(8,12)(9,11)(13,17)(14,16)(15,18)(19,20)(21,22,27,31,26,23,25,29)(24,32,35,28,33,36,34,30);(2,4)(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,23,26,22)(24,28,33,30)(25,29,27,31)(32,34,36,35);(6,11)(7,13)(8,15)(16,20)(21,24)(22,28)(23,30)(25,35)(26,33)(27,34)(29,36)(31,32);(5,8,9,15)(6,12,11,18)(7,14,13,19)(10,16,17,20)(22,23)(24,33)(29,31)(34,35);(7,13)(10,17)(14,19)(16,20)(22,23)(28,30)(29,31)(32,36)Polycyclic presentation:
2^9|a^2=gh,b^2=i,d^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,d^b=di,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6431996);