Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419919 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1195 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,11,16)(7,17,21,23)(8,13,26,35)(9,14,28,18)(10,30,32,15)(12,31,33,20)(19,25,34,24)(22,27,36,29);(1,2)(3,4)(5,7)(6,12)(8,22)(9,13)(10,31)(11,21)(14,24)(15,23)(16,33)(17,30)(18,25)(19,27)(20,32)(26,36)(28,35)(29,34);(2,4)(5,8)(6,13)(7,18)(9,23)(10,24)(11,26)(12,27)(14,21)(15,34)(16,35)(17,28)(19,30)(20,36)(22,31)(25,32)(29,33);(5,9,10,29,11,28,32,27)(6,14,15,36,16,18,30,22)(7,19,31,13,21,34,20,35)(8,23,24,33,26,17,25,12)Polycyclic presentation:
2^9|a^2=i,d^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 335069);