Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23886 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2361 |
| prime radical | #23438 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1625 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,10,17,11,16,30,15)(7,18,23,32,22,33,21,12)(8,24,26,34,20,13,27,35)(9,28,19,36,25,31,29,14);(5,7)(6,12)(8,25)(9,20)(10,23)(11,22)(13,14)(15,33)(16,32)(17,18)(19,27)(21,30)(24,36)(26,29)(28,35)(31,34);(2,4)(5,8)(6,13)(7,19)(9,21)(10,26)(11,20)(12,31)(14,18)(15,34)(16,24)(17,35)(22,29)(23,25)(27,30)(28,32)(33,36);(5,9,10,19,11,25,30,29)(6,14,17,28,16,36,15,31)(7,20,23,27,22,8,21,26)(12,13,18,35,32,24,33,34)Polycyclic presentation:
2^9|a^2=h,d^2=h,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,e^b=ei,g^b=gi,g^c=gi,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 283057);