Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #422540 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2669 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1926 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)(5,6,11,16)(7,17,20,32)(8,21,25,30)(9,15,10,14)(12,26,28,19)(13,24,31,23)(18,29,22,27)(33,34,36,35);(1,2,3,4)(5,7,9,19,11,20,10,26)(6,12,14,17,16,28,15,32)(8,22,23,35,25,18,24,34)(13,29,21,33,31,27,30,36);(5,8,10,23,11,25,9,24)(6,13,15,21,16,31,14,30)(7,18,19,35,20,22,26,34)(12,27,17,33,28,29,32,36);(2,4)(5,7)(6,12)(8,18)(9,26)(10,19)(11,20)(13,27)(14,32)(15,17)(16,28)(21,33)(22,25)(23,35)(24,34)(29,31)(30,36)Polycyclic presentation:
2^9|a^2=i,b^2=e,c^2=f,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^b=gi,e^c=ei,g^c=gi,e^d=ei,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72252198, #72256181, #72256189, … |
| maximal normal subgroup of | #72252198, #72256181, #72256189, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 330141);