Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31282 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2684 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | H(5,𝔽2) |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1030 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)(5,6,12,17,11,18,10,16)(7,19,23,13,22,31,21,32)(8,24,26,35,20,14,28,34)(9,27,25,15,29,33,30,36);(5,7,11,22)(6,13,18,32)(8,25,20,30)(9,28,29,26)(10,23,12,21)(14,33,24,27)(15,35,36,34)(16,31,17,19);(5,8)(6,14,18,24)(7,9)(10,26)(11,20)(12,28)(13,15,32,36)(16,34,17,35)(19,33,31,27)(21,25)(22,29)(23,30);(5,9,11,29)(6,15)(7,20,22,8)(10,30,12,25)(13,24)(14,32)(16,27)(17,33)(18,36)(19,34)(21,26,23,28)(31,35)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=efgi,d^2=efg,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 295278);