Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31282 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2684 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1084 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)(5,6,11,16,10,17,9,15)(7,18,23,12,22,31,21,32)(8,24,26,14,28,13,27,33)(19,30,36,29,20,34,25,35);(5,7,10,22)(6,12,17,32)(8,25,28,36)(9,23,11,21)(13,30,24,34)(14,29,33,35)(15,31,16,18)(19,26,20,27);(5,8)(6,13,17,24)(7,19)(9,26)(10,28)(11,27)(12,29,32,35)(14,15,33,16)(18,34,31,30)(20,22)(21,36)(23,25);(5,8)(6,14,17,33)(7,20)(9,27)(10,28)(11,26)(12,30,32,34)(13,15,24,16)(18,35,31,29)(19,22)(21,36)(23,25)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=efg,d^2=efg,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72286148, #72286161, #72286216, … |
| maximal normal subgroup of | #72286148, #72286161, #72286216, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 295277);