Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9103 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#386 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2754 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)(7,12)(8,20,22,31)(9,14)(10,15)(11,16)(13,21,29,24)(17,32,35,36)(18,26)(19,27)(23,30)(25,34,28,33);(1,2,3,4)(5,7)(6,12)(8,17)(9,18)(10,23)(11,19)(13,28)(14,26)(15,30)(16,27)(20,32)(21,33)(22,35)(24,34)(25,29)(31,36);(5,8)(6,13)(7,17)(9,21)(10,24)(11,22)(12,25)(14,20)(15,31)(16,29)(18,33)(19,35)(23,34)(26,36)(27,28)(30,32);(5,9,11,10)(6,14,16,15)(7,18,19,23)(8,21,22,24)(12,26,27,30)(13,20,29,31)(17,33,35,34)(25,36,28,32)Polycyclic presentation:
2^9|a^2=h,b^2=e,d^2=i,f^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,f^b=fi,f^c=fi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72312492, #72312508, #72312522, … |
| maximal normal subgroup of | #72312492, #72312508, #72312522, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264844);