Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9091 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#386 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1838 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6)(7,12)(8,20,23,13)(9,14)(10,16)(11,15)(17,33,35,26)(18,27)(19,28)(21,36,34,29)(22,32,25,30)(24,31);(1,2,4,3)(5,7)(6,12)(8,21)(9,18)(10,24)(11,19)(13,29)(14,27)(15,28)(16,31)(17,22)(20,36)(23,34)(25,35)(26,30)(32,33);(1,3,4,2)(5,8)(6,13)(7,17)(9,22)(10,25)(11,23)(12,26)(14,30)(15,20)(16,32)(18,34)(19,35)(21,24)(27,36)(28,33)(29,31);(5,9,11,10)(6,14,15,16)(7,18,19,24)(8,22,23,25)(12,27,28,31)(13,30,20,32)(17,34,35,21)(26,36,33,29)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=e,d^2=i,g^2=i,b^a=be,c^a=cf,f^a=fi,c^b=cg,g^b=gi,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72313074, #72313126, #72313180, … |
| maximal normal subgroup of | #72313074, #72313126, #72313180, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265266);