Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24292 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | D4×C4⋊C4 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
862 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,14,11,15,9,25)(7,16,20,27,19,28,17,12)(8,21,24,29,22,35,23,13)(18,32,33,36,34,26,31,30);(1,2,3,4)(5,7)(6,12,15,27)(8,18)(9,17)(10,20)(11,19)(13,26,29,32)(14,16,25,28)(21,30,35,36)(22,34)(23,31)(24,33);(2,4)(7,17,19,20)(8,22)(12,16,27,28)(13,29)(18,33,34,31)(21,35)(23,24)(26,36,32,30);(5,8,9,23,11,22,10,24)(6,13,14,21,15,29,25,35)(7,18,17,31,19,34,20,33)(12,26,16,30,27,32,28,36)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,c^2=f,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,g^b=gi,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 274743);