Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30434 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3005 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1988 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4)(5,6,10,15)(7,16,20,30)(8,12,23,29)(9,13,24,14)(11,18,26,19)(17,32,21,31)(22,27,35,28)(25,34,36,33);(5,7)(6,11)(8,21)(9,18)(10,20)(12,25)(13,16)(14,30)(15,26)(17,23)(19,24)(22,34)(27,31)(28,32)(29,36)(33,35);(5,8,10,23)(6,12,15,29)(7,17,20,21)(9,22,24,35)(11,25,26,36)(13,27,14,28)(16,31,30,32)(18,33,19,34);(6,13)(7,18)(9,24)(12,27)(14,15)(16,30)(17,33)(19,20)(21,34)(22,35)(28,29)(31,32)Polycyclic presentation:
2^9|a^2=h,c^2=i,e^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72284999, #72285534, #72285571, … |
| maximal normal subgroup of | #72284999, #72285534, #72285571, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290343);