Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500443 |
| 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 | C2×#3109 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1938 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(22,26)(24,27)(28,33)(29,34)(30,36)(32,35);(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,22,27,34,25,29,24,26)(23,30,35,33,31,28,32,36);(2,4)(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18)(21,23,25,31)(22,28,29,30)(24,32,27,35)(26,33,34,36);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19)(22,29)(26,34)(28,30)(33,36);(6,12)(10,13)(11,20)(18,19)(22,29)(23,31)(26,34)(32,35)Polycyclic presentation:
2^9|b^2=fi,c^2=h,d^2=hi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=eh,d^c=dhi,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72314897, #72314932, #72314938, … |
| maximal normal subgroup of | #72314897, #72314932, #72314938, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6437061);