Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24283 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C4×#121 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1314 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,15,11,16,9,12)(7,17,20,29,21,26,19,13)(8,22,25,30,23,27,24,14)(18,32,35,36,33,28,34,31);(2,4)(6,12,16,15)(8,23)(13,26,29,17)(14,22,30,27)(18,33)(24,25)(28,31,32,36)(34,35);(1,2,3,4)(5,7)(6,13)(8,18)(9,19)(10,20)(11,21)(12,26)(14,28)(15,17)(16,29)(22,31)(23,33)(24,34)(25,35)(27,36)(30,32);(5,8,9,24,11,23,10,25)(6,14,15,22,16,30,12,27)(7,18,19,34,21,33,20,35)(13,28,17,31,29,32,26,36)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=g,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,d^b=di,f^c=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72239804, #72239813, #72239830, … |
| maximal normal subgroup of | #72239804, #72239813, #72239830, … |
| maximal quotient of | #72265673, C2×#368412 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 275608);