Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #26601 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1810 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,4,3)(8,16)(11,17)(15,20)(18,19)(21,22,23,29,25,30,33,26)(24,34,32,36,35,28,31,27);(2,3)(5,6,9,12)(7,13,14,10)(8,17,16,11)(15,18,20,19)(21,23,25,33)(22,26,30,29)(24,31,35,32)(27,34,36,28);(5,7)(6,10)(8,15)(9,14)(11,18)(12,13)(16,20)(17,19)(21,24)(22,27)(23,31)(25,35)(26,34)(28,29)(30,36)(32,33);(1,3,4,2)(7,14)(10,13)(15,20)(18,19)(21,24)(22,28)(23,32)(25,35)(26,36)(27,29)(30,34)(31,33);(5,8,9,16)(6,11,12,17)(7,15,14,20)(10,18,13,19)(21,23,25,33)(22,29,30,26)(24,31,35,32)(27,28,36,34)Polycyclic presentation:
2^9|a^2=fgi,b^2=hi,d^2=fi,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
| maximal subgroup of | #72280867, #72284013, #72284518, … |
| maximal normal subgroup of | #72280867, #72284013, #72284518, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7091000);