Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393989 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#127 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#399 |
| prime radical | C25 | C2×Q8 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1010 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,9,13,10,14,16,24)(7,15,18,26,19,30,23,11)(8,20,22,34,21,12,27,29)(17,31,33,28,32,36,35,25);(5,7,16,18,10,19,9,23)(6,11,24,15,14,26,13,30)(8,17,27,33,21,32,22,35)(12,28,34,36,20,25,29,31);(5,8)(6,12)(7,17)(9,22)(10,21)(11,25)(13,29)(14,20)(15,31)(16,27)(18,33)(19,32)(23,35)(24,34)(26,28)(30,36);(1,2,3,4)(8,21)(12,20)(17,32)(22,27)(25,28)(29,34)(31,36)(33,35)Polycyclic presentation:
2^9|a^2=e,b^2=ef,d^2=h,e^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,e^b=ei,f^b=fi,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 319968);