Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480274 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C22⋊C4 ◁ C2×#518 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1266 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,11)(8,18,20,28)(9,21,23,13)(10,14)(12,19)(15,29,30,36)(16,25)(17,33,32,26)(22,27)(24,31)(34,35);(1,2,3,4)(5,7)(6,11)(8,15)(9,17)(10,16)(12,24)(13,26)(14,25)(18,29)(19,31)(20,30)(21,33)(22,34)(23,32)(27,35)(28,36);(5,8,10,20)(6,12,14,27)(7,15,16,30)(9,22,23,19)(11,24,25,35)(13,28,21,18)(17,34,32,31)(26,36,33,29);(7,16)(11,25)(15,30)(17,32)(24,35)(26,33)(29,36)(31,34)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=i,f^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^b=di,f^c=fi,g^c=ghi,g^f=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72317932, #72317937, #72317942, … |
| maximal normal subgroup of | #72317932, #72317937, #72317942, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400908);