Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501262 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#447 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ #2929 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1153 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,14,10,16,27,15)(7,17,19,31,21,33,20,12)(8,22,25,13,26,11,23,29)(18,32,34,36,24,30,35,28);(1,2)(3,4)(6,11)(8,23)(9,27)(12,28)(13,14)(15,29)(16,22)(17,30)(18,34)(19,20)(24,35)(25,26)(31,36)(32,33);(2,4)(5,7)(6,12)(8,24)(9,20)(10,21)(11,28)(13,32)(14,33)(15,17)(16,31)(18,26)(19,27)(22,36)(23,35)(25,34)(29,30);(6,11)(8,23)(9,27)(12,28)(13,14)(15,29)(16,22)(17,30)(18,34)(19,20)(24,35)(25,26)(31,36)(32,33)Polycyclic presentation:
2^9|a^2=f,e^2=h,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,e^b=ehi,f^b=fi,h^b=hi,e^c=ei,f^c=fi,e^d=ehi,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72252745 |
| maximal normal subgroup of | #72252745 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408458);