Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #484095 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#509 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1770 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
642 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,15,18,33)(8,20)(9,12)(10,25)(11,17,27,36)(13,23)(14,24)(16,32,35,29)(19,28,26,34)(21,30)(22,31);(5,7)(6,11)(8,16)(9,18)(10,26)(12,27)(13,28)(14,32)(15,31)(17,21)(19,24)(20,33)(22,35)(23,36)(25,29)(30,34);(1,2,3,4)(6,12)(11,27)(13,30)(14,25)(15,33)(20,31)(28,34)(29,32);(7,16)(8,21)(10,24)(11,28)(13,31)(14,25)(15,32)(17,26)(18,35)(19,36)(20,30)(22,23)(27,34)(29,33)Polycyclic presentation:
2^9|a^2=g,c^2=fi,e^2=h,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=de,e^b=ehi,g^b=gi,h^b=hi,e^d=ehi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72239492, #72250863, #72250872, … |
| maximal normal subgroup of | #72239492, #72250863, #72250872, … |
| maximal quotient of | #72287691, #72309433, #72309701, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405062);