Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454002 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3005 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1849 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)(5,6,11,16)(7,17,20,29)(8,21,25,27)(9,12,10,15)(13,26,30,18)(14,23,32,24)(19,28,35,36)(22,33,34,31);(6,12)(7,18)(9,10)(14,27)(15,16)(17,29)(19,34)(20,26)(21,32)(22,35)(23,24)(31,33);(2,4)(5,7)(6,13)(8,22)(9,26)(10,18)(11,20)(12,17)(14,31)(15,29)(16,30)(19,23)(21,36)(24,35)(25,34)(27,28)(32,33);(5,8,11,25)(6,14,16,32)(7,19,20,35)(9,24,10,23)(12,27,15,21)(13,28,30,36)(17,33,29,31)(18,34,26,22)Polycyclic presentation:
2^9|a^2=ef,d^2=i,e^2=i,f^2=i,g^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,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,e^d=ei,f^d=fi,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72238736, #72264996, #72265047, … |
| maximal normal subgroup of | #72238736, #72264996, #72265047, … |
| maximal quotient of | #72238743 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361891);