Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25443 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
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×C22⋊C4 |
| prime radical | #60643 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1836 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)(7,18)(8,21,26,34)(9,12)(10,16)(11,17)(13,25,33,24)(14,20)(15,28)(19,35,36,29)(22,32,23,31)(27,30);(5,7,10,9,11,20,28,27)(6,12,15,18,17,30,16,14)(8,22,25,36,26,23,24,19)(13,31,34,29,33,32,21,35);(1,2,3,4)(5,8,11,26)(6,13,17,33)(7,19,20,36)(9,23,27,22)(10,24,28,25)(12,29,30,35)(14,32,18,31)(15,21,16,34);(5,9,10,20,11,27,28,7)(6,14,15,12,17,18,16,30)(8,23,24,36,26,22,25,19)(13,32,21,29,33,31,34,35)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=fg,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,d^b=di,f^c=fi,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 276368);