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×#393 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2670 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)(8,21,23,29)(9,24)(10,14)(11,15)(12,19)(13,22,30,25)(17,28,33,26)(18,32)(20,27)(31,34,36,35);(2,4)(5,7)(6,12,15,27)(8,17)(9,19)(10,20)(11,18)(13,28,30,26)(14,16,24,32)(21,36,29,31)(22,34)(23,33)(25,35);(5,8)(6,13,15,30)(7,17)(9,25)(10,22)(11,23)(12,26,27,28)(14,21,24,29)(16,31,32,36)(18,33)(19,35)(20,34);(7,18)(12,27)(16,32)(17,33)(19,20)(26,28)(31,36)(34,35)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,e^2=i,f^2=i,b^a=be,c^a=cf,e^a=ei,g^a=gi,c^b=cg,d^b=di,e^c=ei,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72237652, #72237660, #72237670, … |
| maximal normal subgroup of | #72237652, #72237660, #72237670, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265994);