Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419895 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
992 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)(8,21)(9,13,10,15)(12,26)(14,25)(18,28)(19,35,34,27)(20,29)(22,33,36,31)(23,32)(24,30);(5,7,9,20,11,18,10,26)(6,12,15,28,16,29,13,17)(8,22,23,34,24,36,25,19)(14,31,30,27,32,33,21,35);(6,13)(7,18)(9,10)(12,17)(14,30)(15,16)(19,34)(21,32)(23,25)(27,33)(28,29)(31,35);(2,4)(5,8,11,24)(6,14)(7,19,18,34)(9,25,10,23)(12,27)(13,30)(15,21)(16,32)(17,33)(20,36,26,22)(28,31)(29,35)Polycyclic presentation:
2^9|a^2=efg,b^2=f,d^2=ef,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,g^b=gi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72285177, #72285470, #72285552, … |
| maximal normal subgroup of | #72285177, #72285470, #72285552, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 344163);