Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #188252 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3487 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3864 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,16,19,12)(8,20)(9,11)(10,15)(13,21)(14,23)(17,31,33,36)(18,27,34,25)(22,29)(24,28)(26,32,30,35);(1,2)(3,4)(6,11)(9,23)(12,25)(13,20)(14,15)(16,27)(18,34)(21,24)(26,36)(28,29)(30,31)(32,35);(2,4)(5,7)(6,12)(8,17)(9,18)(10,19)(11,25)(13,26)(14,27)(15,16)(20,36)(21,32)(22,33)(23,34)(24,35)(28,30)(29,31);(5,8,10,22)(6,13,15,28)(7,17,19,33)(9,24,23,21)(11,20,14,29)(12,26,16,30)(18,35,34,32)(25,36,27,31)Polycyclic presentation:
2^9|a^2=h,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,h^c=hi,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72286161, #72286995, #72287104, … |
| maximal normal subgroup of | #72286161, #72286995, #72287104, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266836);