Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418863 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2684 |
| 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 | |
1047 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)(5,6,9,14,11,15,10,16)(7,17,19,29,21,12,20,28)(8,22,26,32,24,13,25,31)(18,27,35,30,34,33,23,36);(5,7,11,21)(6,12,15,17)(8,23,24,35)(9,20,10,19)(13,30,22,36)(14,29,16,28)(18,26,34,25)(27,32,33,31);(5,8)(6,13,15,22)(7,18)(9,26)(10,25)(11,24)(12,27,17,33)(14,31,16,32)(19,35)(20,23)(21,34)(28,30,29,36);(2,4)(6,14)(9,10)(12,28)(13,31)(15,16)(17,29)(19,20)(22,32)(23,35)(25,26)(27,30)(33,36)Polycyclic presentation:
2^9|a^2=eg,b^2=i,c^2=ef,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72286161, #72287003, #72287028 |
| maximal normal subgroup of | #72286161, #72287003, #72287028 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 341164);