Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9103 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
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 | C2×#3449 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4446 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)(7,16)(8,20,22,13)(9,14)(10,23)(11,15)(12,18)(17,33,36,32)(19,27)(21,31,25,30)(24,28)(26,34,29,35);(5,7)(6,12)(8,17)(9,18)(10,24)(11,19)(13,29)(14,27)(15,28)(16,23)(20,26)(21,34)(22,36)(25,35)(30,32)(31,33);(2,4)(5,8)(6,13)(7,17)(9,21)(10,25)(11,22)(12,26)(14,30)(15,20)(16,32)(18,34)(19,36)(23,31)(24,35)(27,33)(28,29);(5,9,11,10)(6,14,15,23)(7,18,19,24)(8,21,22,25)(12,27,28,16)(13,30,20,31)(17,34,36,35)(26,33,29,32)Polycyclic presentation:
2^9|a^2=h,d^2=i,e^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,e^c=ei,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72237507, #72237511, #72237694, … |
| maximal normal subgroup of | #72237507, #72237511, #72237694, … |
| maximal quotient of | #72238129, #72257334, #72263952, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264671);