Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533123 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #23573 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2903 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)(3,4)(21,22)(23,27)(24,26)(25,28)(29,32)(30,34)(31,33)(35,36);(5,6)(7,11)(8,12)(9,10)(13,19)(14,17)(15,20)(16,18)(22,26)(24,32)(27,33)(28,29)(31,36)(34,35);(2,4)(6,10)(11,17)(12,18)(19,20)(21,23)(22,27)(24,31)(25,30)(26,33)(28,34)(29,35)(32,36);(5,7,9,14)(6,11,10,17)(8,15,16,13)(12,20,18,19)(23,30)(27,34)(31,36)(33,35);(5,8,9,16)(6,12,10,18)(7,13,14,15)(11,19,17,20)(22,28)(26,29)(27,34)(33,35)Polycyclic presentation:
2^9|d^2=h,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,c^b=ch,f^b=fi,d^c=di,e^d=eh71 conjugacy classes — view the full character table.
| maximal subgroup of | #72308267, #72308294, #72308427 |
| maximal normal subgroup of | #72308267, #72308294, #72308427 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6442283);