Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23472 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#124 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#394 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2658 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)(5,6,10,15,11,14,9,25)(7,12,19,28,17,27,18,35)(8,20,24,31,22,13,23,30)(16,32,34,36,33,26,21,29);(5,7)(6,12)(8,21,22,34)(9,18)(10,19)(11,17)(13,29,20,36)(14,27)(15,28)(16,23,33,24)(25,35)(26,30,32,31);(2,4)(5,8)(6,13)(7,16)(9,24)(10,23)(11,22)(12,26)(14,20)(15,31)(17,33)(18,34)(19,21)(25,30)(27,32)(28,36)(29,35);(7,17)(12,27)(16,33)(18,19)(21,34)(26,32)(28,35)(29,36)Polycyclic presentation:
2^9|a^2=h,b^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fi,g^a=gi,c^b=cg,d^b=di,e^c=ei,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72237172, #72237174, #72237192, … |
| maximal normal subgroup of | #72237172, #72237174, #72237192, … |
| maximal quotient of | #72239007, #72249694, #72257367, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264946);